Some Properties of the Solution to a System of Quaternion Matrix Equations

Abstract

This paper investigates the properties of the $\varphi$-skew-Hermitian solution to the system of quaternion matrix equations involving $\varphi$-skew-Hermicity with four unknowns $A_i{X}_i{\left(A_i\right)}_{\varphi }+B_i{X}_{i+1}{\left(B_i\right)}_{\varphi }=C_i,(i=1,2,3),A_4{X}_4{\left(A_4\right)}_{\varphi }=C_4$. We present the general $\varphi$-skew-Hermitian solution to this system. Moreover, we derive the $\beta \left(\varphi \right)$-signature bounds of the $\varphi$-skew-Hermitian solution $X_1$ in terms of the coefficient matrices. We also give some necessary and sufficient conditions for the system to have $\beta$($\varphi$)-positive semidefinite, $\beta$($\varphi$)-positive definite, $\beta$($\varphi$)-negative semidefinite and $\beta$($\varphi$)-negative definite solutions.

1. Introduction

The quaternion matrix can be used in quantum mechanics [1], color image processing (e.g., [2,3,4]), and signal processing [5], etc. Some researchers have studied the solvability conditions and solutions to some quaternion matrix equations (e.g., [6,7,8,9]).

Hermitian solutions to quaternion matrix equations have been discussed in many papers. Rodman investigated the definitions of $\varphi$-Hermitian, $\varphi$-skew-Hermitian quaternion matrices (Definition 3.6.1 in [10]) and presented a decomposition of $\varphi$-skew-Hermitian quaternion matrix (see Lemma 1). Since then, some researchers have considered the applications of $\varphi$(-skew)-Hermitian quaternion matrices in various aspects. Aghamollaei and Rahjoo [11] established the numerical ranges with respect to nonstandard involutions on quaternionic. Rahjoo et al. [12] studied the numerical ranges with respect to nonstandard involutions. He et al. [13] considered two systems of quaternion matrix equations

$$\left\{\begin{array}{c}{A}_{1}X-Y{B}_1=C_1,\hfill \\ A_{2}Z-Y{B}_2=C_2,\hfill \end{array}Z=Z_{\varphi },\right.$$

and

$$\begin{array}{c}\left\{\begin{array}{c}{A}_{1}X-Y{B}_1=C_1,\hfill \\ A_{2}Y-Z{B}_2=C_2,\hfill \end{array}Z=Z_{\varphi }\right..\hfill \end{array}$$

Wang and Jiang [14] derived the ranks of the skew-Hermitian solution to a classical quaternion matrix equation with two unknowns. The $\eta$-Hermitian quaternion matrix decompositions have applications in signal processing and linear modeling (e.g., [15,16,17,18]). Moreover, He [19] has been investigated the structure, properties and applications of a simultaneous decomposition for quaternion matrices involving $\varphi$-skew-Hermitian. He et al. [20] presented some solvability conditions to a system of quaternion matrix equations involving $\varphi$-skew-Hermicity

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{A}_1{X}_1{\left(A_1\right)}_{\varphi }+B_1{X}_2{\left(B_1\right)}_{\varphi }=C_1,\\ A_2{X}_2{\left(A_2\right)}_{\varphi }+B_2{X}_3{\left(B_2\right)}_{\varphi }=C_2,\\ A_3{X}_3{\left(A_3\right)}_{\varphi }+B_3{X}_4{\left(B_3\right)}_{\varphi }=C_3,\\ A_4{X}_4{\left(A_4\right)}_{\varphi }=C_4,\end{array}\right.X_i=-{\left(X_i\right)}_{\varphi },\end{array}$$

(1)

where $A_i\in {\mathbb{H}}^{p_i\times t_i}$, $B_i\in {\mathbb{H}}^{p_i\times t_{i+1}}$, $C_i\in {\mathbb{H}}^{p_i\times p_i}$, and $C_i$ are $\varphi$-skew-Hermitian matrices. As we know, the solution of the system (1) has not been studied. On the other hand, a special case of the system (1)

$$\begin{array}{c}\hfill A_1{X}_1{\left(A_1\right)}_{\varphi }=C_1\end{array}$$

(2)

can be used in statistics and vibration theory (e.g., [21,22]). The matrix Equation (2) can be used to consider an inverse problem arising in structural modification of the dynamic behaviour (e.g., [23,24]). We conjecture that the main system (1) will also play an important role in the statistics, vibration theory and dynamic behaviour. Inspired by the Hermitian solutions to quaternion matrix equations have widely applications in system and control theory, we consider the expression and properties of the solution to the system (1) in this paper.

The remainder of this paper is organized as follows. In Section 2, we review some definitions and introduce some notations. In Section 3, we provide the general solution to the system (1). In Section 4, we give the $\beta$($\varphi$)-signature bounds of the solution $X_1$ to the system (1) and give some necessary and sufficient conditions for the system (1) to have $\beta$($\varphi$)-positive semidefinite, $\beta$($\varphi$)-positive definite, $\beta$($\varphi$)-negative semidefinite and $\beta$($\varphi$)-negative definite solutions.

2. Preliminaries

In this section, we review some definitions.

Let $\mathbb{R}$ denote the fields of the real numbers. Let $\mathbb{H}$ be a four dimensional vector space over $\mathbb{R}$ with an ordered basis $\left(1,\mathbf{i},\mathbf{j},\mathbf{k}\right)$ [25]. Note that $\mathbf{i},\mathbf{j},\mathbf{k}$ satisfies

$${\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=-1,$$

$$\mathbf{i}\mathbf{j}=-\mathbf{j}\mathbf{i}=\mathbf{k},\mathbf{j}\mathbf{k}=-\mathbf{k}\mathbf{j}=\mathbf{i},\mathbf{k}\mathbf{i}=-\mathbf{i}\mathbf{k}=\mathbf{j}.$$

A real quaternion simply called quaternion is a vector $\mathit{x}=a_0+a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}\in \mathbb{H}$ with real coefficients $a_0,a_1,a_2,a_3$.

The definition of nonstandard involution is giving as follows.

Definition 1

(Non-standard Involution [10]). Let ϕ be an anti-endomorphism of $\mathbb{H}$. Assume that ϕ does not map $\mathbb{H}$ into zero. Then, ϕ is one-to-one and onto $\mathbb{H}$. Thus, ϕ is an anti-automorphism. Moreover, ϕ is real linear and can be represented as a $4\times 4$ real matrix with respect to the basis $\{1,\mathbf{i},\mathbf{j},\mathbf{k}\}$. Then, ϕ is a non-standard involution if and only if

$$\varphi =\left(\begin{array}{cc}1& 0\\ 0& T\end{array}\right),$$

where T is a $3\times 3$ real orthogonal symmetric matrix with eigenvalues $1,1,-1$.

Rodman [10] considered some properties of nonstandard involution. Next, we review the definition of $\varphi$-skew-Hermitian.

Definition 2

($\varphi$-skew-Hermitian [10]). A ∈ ${\mathbb{H}}^{n\times n}$ is said to be ϕ-skew-Hermitian if $A=-{\left(A\right)}_{\varphi }$, where ϕ is a nonstandard involution.

The canonical form of a $\varphi$-skew-Hermitian matrix is presented in [10]. First, we review the definition of $\varphi$-congruent.

Definition 3

($\varphi$-congruent [10]). The quaternion matrices A, B$\in {\mathbb{H}}^{n\times n}$ are ϕ-congruent if $A=SB{S}_{\varphi }$, where S is an invertible quaternion matrix and ϕ is a nonstandard involution.

Lemma 1

([10]). Let ϕ be a nonstandard involution. For every ϕ-skew-Hermitian matrix $A\in {\mathbb{H}}^{n\times n}$, there exists an invertible matrix $S\in {\mathbb{H}}^{n\times n}$ such that

$$\begin{array}{c}\hfill SA{S}_{\varphi }=\left(\begin{array}{ccc}0& 0& 0\\ 0& \beta I_p& 0\\ 0& 0& -\beta I_q\end{array}\right),\beta =\beta \left(\varphi \right),\end{array}$$

(3)

where the unit ϕ-skew-Hermitian quaternion β is fixed and denoted by $\beta \left(\varphi \right)$. Moreover, the integers p and q are uniquely determined by A (for a fixed β(ϕ)).

According to Lemma 1, the definition of $\beta \left(\varphi \right)$-signature of a $\varphi$-skew-Hermitian quaternion matrix A is provided.

Definition 4

($\beta \left(\varphi \right)$-signature [10]). We say that the ordered triple of nonnegative integers

$$({ln}_{+}\left(A\right),{ln}_{-}\left(A\right),{ln}_0\left(A\right)):=(p,q,n-p-q)$$

is the $\beta \left(\varphi \right)$-signature of a ϕ-skew-Hermitian quaternion matrix A, as in Lemma 1. The matrix A is said to be $\beta \left(\varphi \right)$-positive definite, $\beta \left(\varphi \right)$-positive semidefinite, if ${ln}_{+}\left(A\right)=n$, ${ln}_{+}\left(A\right)+{ln}_0\left(A\right)=n$, respectively. Analogously, $\beta \left(\varphi \right)$-negative definite and $\beta \left(\varphi \right)$-negative semidefinite ϕ-skew-Hermitian quaternion matrices are defined.

3. The General ϕ-Skew-Hermitian Solution to the System (1)

In this section, we provide the general $\varphi$-skew-Hermitian solution to the system (1).

Using the results of Lemma 1 in [20], there exist nonsingular matrices $\widehat{T_i}\in {\mathbb{H}}^{t_i\times t_i},$$\widehat{P_i}\in {\mathbb{H}}^{p_i\times p_i},(i=1,2,3)$, ${\widehat{T}}_4\in {\mathbb{H}}^{t_4\times t_4},{\widehat{P}}_4\in {\mathbb{H}}^{p_4\times p_4}$ such that

$$\begin{array}{c}\hfill \widehat{P_i}{A}_i\widehat{T_i}=S_{a_i},\widehat{P_i}{B}_i{\widehat{T}}_{i+1}=S_{b_i},{\widehat{P}}_4{A}_4{\widehat{T}}_4=S_{a_4}.\end{array}$$

Therefore, the system (1) is equivalent to the following system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{S}_{a_1}{\widehat{X}}_1{\left(S_{a_1}\right)}_{\varphi }+S_{b_1}{\widehat{X}}_2{\left(S_{b_1}\right)}_{\varphi }=D_{kj}^{\left(1\right)},\\ S_{a_2}{\widehat{X}}_2{\left(S_{a_2}\right)}_{\varphi }+S_{b_2}{\widehat{X}}_3{\left(S_{b_2}\right)}_{\varphi }=D_{kj}^{\left(2\right)},\\ S_{a_3}{\widehat{X}}_3{\left(S_{a_3}\right)}_{\varphi }+S_{b_3}{\widehat{X}}_4{\left(S_{b_3}\right)}_{\varphi }=D_{kj}^{\left(3\right)},\\ S_{a_4}{\widehat{X}}_4{\left(S_{a_4}\right)}_{\varphi }=D_{kj}^{\left(4\right)},\end{array}\right.\end{array}$$

(4)

where $X_i=-{\left(X_i\right)}_{\varphi },D_{kj}^{\left(i\right)}={\widehat{P}}_i{C}_i{\left({\widehat{P}}_i\right)}_{\varphi },$$S_{a_i}$ and $S_{b_i}$ have the following form

$$\begin{array}{cc}\hfill & \left(\begin{array}{cccc}{S}_{a_1}& S_{b_1}& & \\ & S_{a_2}& S_{b_2}& \\ & & S_{a_3}& S_{b_3}\\ & & & S_{a_4}\end{array}\right)\hfill \\ & =\left(\begin{array}{cccc}\begin{array}{c}\begin{array}{cccccccc}I& 0& 0& 0& 0& 0& 0& 0\\ 0& I& 0& 0& 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0& 0& 0\\ 0& 0& 0& I& 0& 0& 0& 0\\ 0& 0& 0& 0& I& 0& 0& 0\\ 0& 0& 0& 0& 0& I& 0& 0\\ 0& 0& 0& 0& 0& 0& I& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}& \begin{array}{c}\begin{array}{cccccccccccccccccc}I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}& & \\ & \begin{array}{c}\begin{array}{cccccccccccccccccc}I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}& \begin{array}{cc}\begin{array}{cccccccccccccccccccc}I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& I& 0& 0& 0& 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0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}& \begin{array}{c}\begin{array}{cccccccccccccc}I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}\\ & & & \begin{array}{c}\begin{array}{cccccccccccccc}I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}\end{array}\right).\hfill \end{array}$$

(5)

The above idea and symbols are presented in [20].

In order to give the general $\varphi$-skew-Hermitian solution to the system (1), we need to obtain the general $\varphi$-skew-Hermitian solution to the system (4). The following theorem gives the general $\varphi$-skew-Hermitian solution to the system (1).

Theorem 1.

Assume that the system (1) is consistent. The general ϕ-skew-Hermitian solution to the system (1) can be expressed as

$$\begin{array}{c}\hfill X_1={\widehat{T}}_1{\widehat{X}}_1{\left({\widehat{T}}_1\right)}_{\varphi },X_2={\widehat{T}}_2{\widehat{X}}_2{\left({\widehat{T}}_2\right)}_{\varphi },X_3={\widehat{T}}_3{\widehat{X}}_3{\left({\widehat{T}}_3\right)}_{\varphi },X_4={\widehat{T}}_4{\widehat{X}}_4{\left({\widehat{T}}_4\right)}_{\varphi },\end{array}$$

(6)

where

$$\begin{array}{c}\hfill {\widehat{X}}_1=\left(\begin{array}{ccc}{D}_{11}^{\left(1\right)}-D_{11}^{\left(2\right)}+D_{11}^{\left(3\right)}-D_{11}^{\left(4\right)}& D_{12}^{\left(1\right)}-D_{12}^{\left(2\right)}+D_{12}^{\left(3\right)}-X_{12}^{\left(4\right)}& D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)}\\ -{(D_{12}^{\left(1\right)}-D_{12}^{\left(2\right)}+D_{12}^{\left(3\right)}-X_{12}^{\left(4\right)})}_{\varphi }& D_{22}^{\left(1\right)}-D_{22}^{\left(2\right)}+D_{22}^{\left(3\right)}-X_{22}^{\left(4\right)}& D_{23}^{\left(1\right)}-D_{23}^{\left(2\right)}+D_{23}^{\left(3\right)}\\ -{(D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)})}_{\varphi }& -{(D_{23}^{\left(1\right)}-D_{23}^{\left(2\right)}+D_{23}^{\left(3\right)})}_{\varphi }& D_{33}^{\left(1\right)}-D_{33}^{\left(2\right)}+D_{33}^{\left(3\right)}\\ -{(D_{14}^{\left(1\right)}-D_{14}^{\left(2\right)}+X_{14}^{\left(3\right)})}_{\varphi }& -{(D_{24}^{\left(1\right)}-D_{24}^{\left(2\right)}+X_{24}^{\left(3\right)})}_{\varphi }& -{(D_{34}^{\left(1\right)}-D_{34}^{\left(2\right)}+X_{34}^{\left(3\right)})}_{\varphi }\\ -{(D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)})}_{\varphi }& -{(D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)})}_{\varphi }& -{(D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)})}_{\varphi }\\ -{(D_{16}^{\left(1\right)}-X_{16}^{\left(2\right)})}_{\varphi }& -{(D_{26}^{\left(1\right)}-X_{26}^{\left(2\right)})}_{\varphi }& -{(D_{36}^{\left(1\right)}-X_{36}^{\left(2\right)})}_{\varphi }\\ -{\left(D_{17}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{27}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{37}^{\left(1\right)}\right)}_{\varphi }\\ -{\left(X_{18}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{28}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{38}^{\left(1\right)}\right)}_{\varphi }\end{array}\right.\\ \hfill \left.\begin{array}{ccccc}{D}_{14}^{\left(1\right)}-D_{14}^{\left(2\right)}+X_{14}^{\left(3\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}& D_{16}^{\left(1\right)}-X_{16}^{\left(2\right)}& D_{17}^{\left(1\right)}& X_{18}^{\left(1\right)}\\ D_{24}^{\left(1\right)}-D_{24}^{\left(2\right)}+X_{24}^{\left(3\right)}& D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)}& D_{26}^{\left(1\right)}-X_{26}^{\left(2\right)}& D_{27}^{\left(1\right)}& X_{28}^{\left(1\right)}\\ D_{34}^{\left(1\right)}-D_{34}^{\left(2\right)}+X_{34}^{\left(3\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}& D_{36}^{\left(1\right)}-X_{36}^{\left(2\right)}& D_{37}^{\left(1\right)}& X_{38}^{\left(1\right)}\\ D_{44}^{\left(1\right)}-D_{44}^{\left(2\right)}+X_{44}^{\left(3\right)}& D_{45}^{\left(1\right)}-D_{45}^{\left(2\right)}& D_{46}^{\left(1\right)}-X_{46}^{\left(2\right)}& D_{47}^{\left(1\right)}& X_{48}^{\left(1\right)}\\ -{(D_{45}^{\left(1\right)}-D_{45}^{\left(2\right)})}_{\varphi }& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}& D_{56}^{\left(1\right)}-X_{56}^{\left(2\right)}& D_{57}^{\left(1\right)}& X_{58}^{\left(1\right)}\\ -{(D_{46}^{\left(1\right)}-X_{46}^{\left(2\right)})}_{\varphi }& -{(D_{56}^{\left(1\right)}-X_{56}^{\left(2\right)})}_{\varphi }& D_{66}^{\left(1\right)}-X_{66}^{\left(2\right)}& D_{67}^{\left(1\right)}& X_{68}^{\left(1\right)}\\ -{\left(D_{47}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{67}^{\left(1\right)}\right)}_{\varphi }& D_{77}^{\left(1\right)}& X_{78}^{\left(1\right)}\\ -{\left(X_{48}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{58}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{68}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{78}^{\left(1\right)}\right)}_{\varphi }& X_{88}^{\left(1\right)}\end{array}\right),\end{array}$$

(7)

$$\begin{array}{c}\hfill {\widehat{X}}_2=\left(\begin{array}{cccccc}{D}_{11}^{\left(2\right)}-D_{11}^{\left(3\right)}+D_{11}^{\left(4\right)}& D_{12}^{\left(2\right)}-D_{12}^{\left(3\right)}+X_{12}^{\left(4\right)}& D_{13}^{\left(2\right)}-D_{13}^{\left(3\right)}& D_{14}^{\left(2\right)}-X_{14}^{\left(3\right)}& D_{15}^{\left(2\right)}& X_{16}^{\left(2\right)}\\ -{(D_{12}^{\left(2\right)}-D_{12}^{\left(3\right)}+X_{12}^{\left(4\right)})}_{\varphi }& D_{22}^{\left(2\right)}-D_{22}^{\left(3\right)}+X_{22}^{\left(4\right)}& D_{23}^{\left(2\right)}-D_{23}^{\left(3\right)}& D_{24}^{\left(2\right)}-X_{24}^{\left(3\right)}& D_{25}^{\left(2\right)}& X_{26}^{\left(2\right)}\\ -{(D_{13}^{\left(2\right)}-D_{13}^{\left(3\right)})}_{\varphi }& -{(D_{23}^{\left(2\right)}-D_{23}^{\left(3\right)})}_{\varphi }& D_{33}^{\left(2\right)}-D_{33}^{\left(3\right)}& D_{34}^{\left(2\right)}-X_{34}^{\left(3\right)}& D_{35}^{\left(2\right)}& X_{36}^{\left(2\right)}\\ -{(D_{14}^{\left(2\right)}-X_{14}^{\left(3\right)})}_{\varphi }& -{(D_{24}^{\left(2\right)}-X_{24}^{\left(3\right)})}_{\varphi }& -{(D_{34}^{\left(2\right)}-X_{34}^{\left(3\right)})}_{\varphi }& D_{44}^{\left(2\right)}-X_{44}^{\left(3\right)}& D_{45}^{\left(2\right)}& X_{46}^{\left(2\right)}\\ -{\left(D_{15}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{25}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{35}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{45}^{\left(2\right)}\right)}_{\varphi }& D_{55}^{\left(2\right)}& X_{56}^{\left(2\right)}\\ -{\left(X_{16}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{26}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{36}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{46}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{56}^{\left(2\right)}\right)}_{\varphi }& X_{66}^{\left(2\right)}\\ -{(D_{16}^{\left(2\right)}-D_{14}^{\left(3\right)}+D_{12}^{\left(4\right)})}_{\varphi }& -{(D_{26}^{\left(2\right)}-D_{24}^{\left(3\right)}+X_{23}^{\left(4\right)})}_{\varphi }& -{(D_{36}^{\left(2\right)}-D_{34}^{\left(3\right)})}_{\varphi }& -{(D_{46}^{\left(2\right)}-X_{45}^{\left(3\right)})}_{\varphi }& -{\left(D_{56}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{68}^{\left(1\right)}\right)}_{\varphi }\\ -{(D_{17}^{\left(2\right)}-D_{15}^{\left(3\right)}+X_{14}^{\left(4\right)})}_{\varphi }& -{(D_{27}^{\left(2\right)}-D_{25}^{\left(3\right)}+X_{24}^{\left(4\right)})}_{\varphi }& -{(D_{37}^{\left(2\right)}-D_{35}^{\left(3\right)})}_{\varphi }& -{(D_{47}^{\left(2\right)}-X_{46}^{\left(3\right)})}_{\varphi }& -{\left(D_{57}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{69}^{\left(1\right)}\right)}_{\varphi }\\ -{(D_{18}^{\left(2\right)}-D_{16}^{\left(3\right)})}_{\varphi }& -{(D_{28}^{\left(2\right)}-D_{26}^{\left(3\right)})}_{\varphi }& -{(D_{38}^{\left(2\right)}-D_{36}^{\left(3\right)})}_{\varphi }& -{(D_{48}^{\left(2\right)}-X_{47}^{\left(3\right)})}_{\varphi }& -{\left(D_{58}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{6,10}^{\left(1\right)}\right)}_{\varphi }\\ -{(D_{19}^{\left(2\right)}-X_{18}^{\left(3\right)})}_{\varphi }& -{(D_{29}^{\left(2\right)}-X_{28}^{\left(3\right)})}_{\varphi }& -{(D_{39}^{\left(2\right)}-X_{38}^{\left(3\right)})}_{\varphi }& -{(D_{49}^{\left(2\right)}-X_{48}^{\left(3\right)})}_{\varphi }& -{\left(D_{59}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{6,11}^{\left(1\right)}\right)}_{\varphi }\\ -{\left(D_{1,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{2,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{3,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{4,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{5,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{6,12}^{\left(1\right)}\right)}_{\varphi }\\ -{\left(D_{1,13}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{2,13}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{3,13}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{4,13}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{5,13}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{6,13}^{\left(1\right)}\right)}_{\varphi }\\ -{(D_{1,11}^{\left(2\right)}-D_{17}^{\left(3\right)}+D_{13}^{\left(4\right)})}_{\varphi }& -{(D_{2,11}^{\left(2\right)}-D_{27}^{\left(3\right)}+X_{25}^{\left(4\right)})}_{\varphi }& -{(D_{3,11}^{\left(2\right)}-D_{37}^{\left(3\right)})}_{\varphi }& -{(D_{4,11}^{\left(2\right)}-X_{49}^{\left(3\right)})}_{\varphi }& -{\left(D_{5,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{6,13}^{\left(2\right)}\right)}_{\varphi }\\ -{(D_{1,12}^{\left(2\right)}-D_{18}^{\left(3\right)}+X_{16}^{\left(4\right)})}_{\varphi }& -{(D_{2,12}^{\left(2\right)}-D_{28}^{\left(3\right)}+X_{26}^{\left(4\right)})}_{\varphi }& -{(D_{3,12}^{\left(2\right)}-D_{38}^{\left(3\right)})}_{\varphi }& -{(D_{4,12}^{\left(2\right)}-X_{4,10}^{\left(3\right)})}_{\varphi }& -{\left(D_{5,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{6,14}^{\left(2\right)}\right)}_{\varphi }\\ -{(D_{1,13}^{\left(2\right)}-D_{19}^{\left(3\right)})}_{\varphi }& -{(D_{2,13}^{\left(2\right)}-D_{29}^{\left(3\right)})}_{\varphi }& -{(D_{3,13}^{\left(2\right)}-D_{39}^{\left(3\right)})}_{\varphi }& -{(D_{4,13}^{\left(2\right)}-X_{4,11}^{\left(3\right)})}_{\varphi }& -{\left(D_{5,13}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{6,15}^{\left(2\right)}\right)}_{\varphi }\\ -{(D_{1,14}^{\left(2\right)}-X_{1,12}^{\left(3\right)})}_{\varphi }& -{(D_{2,14}^{\left(2\right)}-X_{2,12}^{\left(3\right)})}_{\varphi }& -{(D_{3,14}^{\left(2\right)}-X_{3,12}^{\left(3\right)})}_{\varphi }& -{(D_{4,14}^{\left(2\right)}-X_{4,12}^{\left(3\right)})}_{\varphi }& -{\left(D_{5,14}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{6,16}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(D_{1,15}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{2,15}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{3,15}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{4,15}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{5,15}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{6,17}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{1,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{2,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{3,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{4,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{5,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{6,18}^{\left(2\right)}\right)}_{\varphi }\end{array}\right.\\ \hfill \begin{array}{cccccc}{D}_{16}^{\left(2\right)}-D_{14}^{\left(3\right)}+D_{12}^{\left(4\right)}& D_{17}^{\left(2\right)}-D_{15}^{\left(3\right)}+X_{14}^{\left(4\right)}& D_{18}^{\left(2\right)}-D_{16}^{\left(3\right)}& D_{19}^{\left(2\right)}-X_{18}^{\left(3\right)}& D_{1,10}^{\left(2\right)}& D_{1,13}^{\left(1\right)}\\ D_{26}^{\left(2\right)}-D_{24}^{\left(3\right)}+X_{23}^{\left(4\right)}& D_{27}^{\left(2\right)}-D_{25}^{\left(3\right)}+X_{24}^{\left(4\right)}& D_{28}^{\left(2\right)}-D_{26}^{\left(3\right)}& D_{29}^{\left(2\right)}-X_{28}^{\left(3\right)}& D_{2,10}^{\left(2\right)}& D_{2,13}^{\left(1\right)}\\ D_{36}^{\left(2\right)}-D_{34}^{\left(3\right)}& D_{37}^{\left(2\right)}-D_{35}^{\left(3\right)}& D_{38}^{\left(2\right)}-D_{36}^{\left(3\right)}& D_{39}^{\left(2\right)}-X_{38}^{\left(3\right)}& D_{3,10}^{\left(2\right)}& D_{3,13}^{\left(1\right)}\\ D_{46}^{\left(2\right)}-X_{45}^{\left(3\right)}& D_{47}^{\left(2\right)}-X_{46}^{\left(3\right)}& D_{48}^{\left(2\right)}-X_{47}^{\left(3\right)}& D_{49}^{\left(2\right)}-X_{48}^{\left(3\right)}& D_{4,10}^{\left(2\right)}& D_{4,13}^{\left(1\right)}\\ D_{56}^{\left(2\right)}& D_{57}^{\left(2\right)}& D_{58}^{\left(2\right)}& D_{59}^{\left(2\right)}& D_{5,10}^{\left(2\right)}& D_{5,13}^{\left(1\right)}\\ D_{68}^{\left(1\right)}& D_{69}^{\left(1\right)}& D_{6,10}^{\left(1\right)}& D_{6,11}^{\left(1\right)}& D_{6,12}^{\left(1\right)}& D_{6,13}^{\left(1\right)}\\ D_{66}^{\left(2\right)}-D_{44}^{\left(3\right)}+D_{22}^{\left(4\right)}& D_{67}^{\left(2\right)}-D_{45}^{\left(3\right)}+X_{34}^{\left(4\right)}& D_{68}^{\left(2\right)}-D_{46}^{\left(3\right)}& D_{69}^{\left(2\right)}-X_{58}^{\left(3\right)}& D_{6,10}^{\left(2\right)}& D_{8,13}^{\left(1\right)}\\ -{(D_{67}^{\left(2\right)}-D_{45}^{\left(3\right)}+X_{34}^{\left(4\right)})}_{\varphi }& D_{77}^{\left(2\right)}-D_{55}^{\left(3\right)}+X_{44}^{\left(4\right)}& D_{78}^{\left(2\right)}-D_{56}^{\left(3\right)}& D_{79}^{\left(2\right)}-X_{68}^{\left(3\right)}& D_{7,10}^{\left(2\right)}& D_{9,13}^{\left(1\right)}\\ -{(D_{68}^{\left(2\right)}-D_{46}^{\left(3\right)})}_{\varphi }& -{(D_{78}^{\left(2\right)}-D_{56}^{\left(3\right)})}_{\varphi }& D_{88}^{\left(2\right)}-D_{66}^{\left(3\right)}& D_{89}^{\left(2\right)}-X_{78}^{\left(3\right)}& D_{8,10}^{\left(2\right)}& D_{10,13}^{\left(1\right)}\\ -{(D_{69}^{\left(2\right)}-X_{58}^{\left(3\right)})}_{\varphi }& -{(D_{79}^{\left(2\right)}-X_{68}^{\left(3\right)})}_{\varphi }& -{(D_{89}^{\left(2\right)}-X_{78}^{\left(3\right)})}_{\varphi }& D_{99}^{\left(2\right)}-X_{88}^{\left(3\right)}& D_{9,10}^{\left(2\right)}& D_{11,13}^{\left(1\right)}\\ -{\left(D_{6,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{7,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{8,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{9,10}^{\left(2\right)}\right)}_{\varphi }& D_{10,10}^{\left(2\right)}& D_{12,13}^{\left(1\right)}\\ -{\left(D_{8,13}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{9,13}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{10,13}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{11,13}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{12,13}^{\left(1\right)}\right)}_{\varphi }& D_{13,13}^{\left(1\right)}\\ -{(D_{6,11}^{\left(2\right)}-D_{47}^{\left(3\right)}+D_{23}^{\left(4\right)})}_{\varphi }& -{(D_{7,11}^{\left(2\right)}-D_{57}^{\left(3\right)}+X_{45}^{\left(4\right)})}_{\varphi }& -{(D_{8,11}^{\left(2\right)}-X_{69}^{\left(3\right)})}_{\varphi }& -(D_{9,11}^{\left(2\right)}-X_{89}^{\left(3\right)})\varphi & -\left(D_{10,11}^{\left(2\right)}\right)\varphi & -{\left(X_{12,13}^{\left(2\right)}\right)}_{\varphi }\\ -{(D_{6,12}^{\left(2\right)}-D_{48}^{\left(3\right)}+X_{36}^{\left(4\right)})}_{\varphi }& -{(D_{7,12}^{\left(2\right)}-D_{58}^{\left(3\right)}+X_{46}^{\left(4\right)})}_{\varphi }& -{(D_{8,12}^{\left(2\right)}-D_{68}^{\left(3\right)})}_{\varphi }& -{(D_{9,12}^{\left(2\right)}-X_{8,10}^{\left(3\right)})}_{\varphi }& -{\left(D_{10,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{12,14}^{\left(2\right)}\right)}_{\varphi }\\ -{(D_{6,13}^{\left(2\right)}-D_{49}^{\left(3\right)})}_{\varphi }& -{(D_{7,13}^{\left(2\right)}-D_{59}^{\left(3\right)})}_{\varphi }& -{(D_{8,13}^{\left(2\right)}-D_{69}^{\left(3\right)})}_{\varphi }& -{(D_{9,13}^{\left(2\right)}-X_{8,11}^{\left(3\right)})}_{\varphi }& -{\left(D_{10,13}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{12,15}^{\left(2\right)}\right)}_{\varphi }\\ -{(D_{6,14}^{\left(2\right)}-X_{5,12}^{\left(3\right)})}_{\varphi }& -{(D_{7,14}^{\left(2\right)}-X_{6,12}^{\left(3\right)})}_{\varphi }& -{(D_{8,14}^{\left(2\right)}-X_{7,12}^{\left(3\right)})}_{\varphi }& -{(D_{9,14}^{\left(2\right)}-X_{8,12}^{\left(3\right)})}_{\varphi }& -{\left(D_{10,14}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{12,16}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(D_{6,15}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{7,15}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{8,15}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{9,15}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{10,15}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{12,17}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{7,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{8,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{9,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{10,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{11,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{12,18}^{\left(2\right)}\right)}_{\varphi }\end{array}\\ \hfill \left.\begin{array}{cccccc}{D}_{1,11}^{\left(2\right)}-D_{17}^{\left(3\right)}+D_{13}^{\left(4\right)}& D_{1,12}^{\left(2\right)}-D_{18}^{\left(3\right)}+X_{16}^{\left(4\right)}& D_{1,13}^{\left(2\right)}-D_{19}^{\left(3\right)}& D_{1,14}^{\left(2\right)}-X_{1,12}^{\left(3\right)}& D_{1,15}^{\left(2\right)}& X_{1,18}^{\left(2\right)}\\ D_{2,11}^{\left(2\right)}-D_{27}^{\left(3\right)}+X_{25}^{\left(4\right)}& D_{2,12}^{\left(2\right)}-D_{28}^{\left(3\right)}+X_{26}^{\left(4\right)}& D_{2,13}^{\left(2\right)}-D_{29}^{\left(3\right)}& D_{2,14}^{\left(2\right)}-X_{2,12}^{\left(3\right)}& D_{2,15}^{\left(2\right)}& X_{2,18}^{\left(2\right)}\\ D_{3,11}^{\left(2\right)}-D_{37}^{\left(3\right)}& D_{3,12}^{\left(2\right)}-D_{38}^{\left(3\right)}& D_{3,13}^{\left(2\right)}-D_{39}^{\left(3\right)}& D_{3,14}^{\left(2\right)}-X_{3,12}^{\left(3\right)}& D_{3,15}^{\left(2\right)}& X_{3,18}^{\left(2\right)}\\ D_{4,11}^{\left(2\right)}-X_{49}^{\left(3\right)}& D_{4,12}^{\left(2\right)}-X_{4,10}^{\left(3\right)}& D_{4,13}^{\left(2\right)}-X_{4,11}^{\left(3\right)}& D_{4,14}^{\left(2\right)}-X_{4,12}^{\left(3\right)}& D_{4,15}^{\left(2\right)}& X_{4,18}^{\left(2\right)}\\ D_{5,11}^{\left(2\right)}& D_{5,12}^{\left(2\right)}& D_{5,13}^{\left(2\right)}& D_{5,14}^{\left(2\right)}& D_{5,15}^{\left(2\right)}& X_{5,18}^{\left(2\right)}\\ X_{6,13}^{\left(2\right)}& X_{6,14}^{\left(2\right)}& X_{6,15}^{\left(2\right)}& X_{6,16}^{\left(2\right)}& X_{6,17}^{\left(2\right)}& X_{6,18}^{\left(2\right)}\\ D_{6,11}^{\left(2\right)}-D_{47}^{\left(3\right)}+D_{23}^{\left(4\right)}& D_{6,12}^{\left(2\right)}-D_{48}^{\left(3\right)}+X_{36}^{\left(4\right)}& D_{6,13}^{\left(2\right)}-D_{49}^{\left(3\right)}& D_{6,14}^{\left(2\right)}-X_{5,12}^{\left(3\right)}& D_{6,15}^{\left(2\right)}& X_{7,18}^{\left(2\right)}\\ D_{7,11}^{\left(2\right)}-D_{57}^{\left(3\right)}+X_{45}^{\left(4\right)}& D_{7,12}^{\left(2\right)}-D_{58}^{\left(3\right)}+X_{46}^{\left(4\right)}& D_{7,13}^{\left(2\right)}-D_{59}^{\left(3\right)}& D_{7,14}^{\left(2\right)}-X_{6,12}^{\left(3\right)}& D_{7,15}^{\left(2\right)}& X_{8,18}^{\left(2\right)}\\ D_{8,11}^{\left(2\right)}-D_{67}^{\left(3\right)}& D_{8,12}^{\left(2\right)}-D_{68}^{\left(3\right)}& D_{8,13}^{\left(2\right)}-D_{69}^{\left(3\right)}& D_{8,14}^{\left(2\right)}-X_{7,12}^{\left(3\right)}& D_{8,15}^{\left(2\right)}& X_{9,18}^{\left(2\right)}\\ D_{9,11}^{\left(2\right)}-X_{89}^{\left(3\right)}& D_{9,12}^{\left(2\right)}-X_{8,10}^{\left(3\right)}& D_{9,13}^{\left(2\right)}-X_{8,11}^{\left(3\right)}& D_{9,14}^{\left(2\right)}-X_{8,12}^{\left(3\right)}& D_{9,15}^{\left(2\right)}& X_{10,18}^{\left(2\right)}\\ D_{10,11}^{\left(2\right)}& D_{10,12}^{\left(2\right)}& D_{10,13}^{\left(2\right)}& D_{10,14}^{\left(2\right)}& D_{10,15}^{\left(2\right)}& X_{11,18}^{\left(2\right)}\\ X_{12,13}^{\left(2\right)}& X_{12,14}^{\left(2\right)}& X_{12,15}^{\left(2\right)}& X_{12,16}^{\left(2\right)}& X_{12,17}^{\left(2\right)}& X_{12,18}^{\left(2\right)}\\ D_{11,11}^{\left(2\right)}-D_{77}^{\left(3\right)}+D_{33}^{\left(4\right)}& D_{11,12}^{\left(2\right)}-D_{78}^{\left(3\right)}+X_{56}^{\left(4\right)}& D_{11,13}^{\left(2\right)}-D_{79}^{\left(3\right)}& D_{11,14}^{\left(2\right)}-X_{9,12}^{\left(3\right)}& D_{11,15}^{\left(2\right)}& X_{13,18}^{\left(2\right)}\\ -{(D_{11,12}^{\left(2\right)}-D_{78}^{\left(3\right)}+X_{56}^{\left(4\right)})}_{\varphi }& D_{12,12}^{\left(2\right)}-D_{88}^{\left(3\right)}+X_{66}^{\left(4\right)}& D_{12,13}^{\left(2\right)}-X_{10,11}^{\left(3\right)}& D_{12,14}^{\left(2\right)}-X_{10,12}^{\left(3\right)}& D_{12,15}^{\left(2\right)}& X_{14,18}^{\left(2\right)}\\ -{(D_{11,13}^{\left(2\right)}-D_{79}^{\left(3\right)})}_{\varphi }& -{(D_{12,13}^{\left(2\right)}-X_{10,11}^{\left(3\right)})}_{\varphi }& D_{13,13}^{\left(2\right)}-D_{99}^{\left(3\right)}& D_{13,14}^{\left(2\right)}-X_{11,12}^{\left(3\right)}& D_{13,15}^{\left(2\right)}& X_{15,18}^{\left(2\right)}\\ -{(D_{11,14}^{\left(2\right)}-X_{9,12}^{\left(3\right)})}_{\varphi }& -{(D_{12,14}^{\left(2\right)}-X_{10,12}^{\left(3\right)})}_{\varphi }& -{(D_{13,14}^{\left(2\right)}-X_{11,12}^{\left(3\right)})}_{\varphi }& D_{14,14}^{\left(2\right)}-X_{12,12}^{\left(3\right)}& D_{14,15}^{\left(2\right)}& X_{16,18}^{\left(2\right)}\\ -{\left(D_{11,15}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{12,15}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{13,15}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{14,15}^{\left(2\right)}\right)}_{\varphi }& D_{15,15}^{\left(2\right)}& X_{17,18}^{\left(2\right)}\\ -{\left(X_{13,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{14,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{15,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{16,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{17,18}^{\left(2\right)}\right)}_{\varphi }& X_{18,18}^{\left(2\right)}\end{array}\right),\end{array}$$

(8)

$$\begin{array}{c}\hfill {\widehat{X}}_3=\left(\begin{array}{ccccccc}{D}_{11}^{\left(3\right)}-D_{11}^{\left(4\right)}& D_{12}^{\left(3\right)}-X_{12}^{\left(4\right)}& D_{13}^{\left(3\right)}& X_{14}^{\left(3\right)}& D_{14}^{\left(3\right)}-D_{12}^{\left(4\right)}& D_{15}^{\left(3\right)}-X_{14}^{\left(4\right)}& D_{16}^{\left(3\right)}\\ -{(D_{12}^{\left(3\right)}-X_{12}^{\left(4\right)})}_{\varphi }& D_{22}^{\left(3\right)}-X_{22}^{\left(4\right)}& D_{23}^{\left(3\right)}& X_{24}^{\left(3\right)}& D_{24}^{\left(3\right)}-X_{23}^{\left(4\right)}& D_{25}^{\left(3\right)}-X_{24}^{\left(4\right)}& D_{26}^{\left(3\right)}\\ -{\left(D_{13}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{23}^{\left(3\right)}\right)}_{\varphi }& D_{33}^{\left(3\right)}& X_{34}^{\left(3\right)}& D_{34}^{\left(3\right)}& D_{35}^{\left(3\right)}& D_{36}^{\left(3\right)}\\ -{\left(X_{14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{24}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{34}^{\left(3\right)}\right)}_{\varphi }& X_{44}^{\left(3\right)}& X_{45}^{\left(3\right)}& X_{46}^{\left(3\right)}& X_{47}^{\left(3\right)}\\ -{(D_{14}^{\left(3\right)}-D_{12}^{\left(4\right)})}_{\varphi }& -{(D_{24}^{\left(3\right)}-X_{23}^{\left(4\right)})}_{\varphi }& -{\left(D_{34}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{45}^{\left(3\right)}\right)}_{\varphi }& D_{44}^{\left(3\right)}-D_{22}^{\left(4\right)}& D_{45}^{\left(3\right)}-X_{34}^{\left(4\right)}& D_{46}^{\left(3\right)}\\ -{(D_{15}^{\left(3\right)}-X_{14}^{\left(4\right)})}_{\varphi }& -{(D_{25}^{\left(3\right)}-X_{24}^{\left(4\right)})}_{\varphi }& -{\left(D_{35}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{46}^{\left(3\right)}\right)}_{\varphi }& -{(D_{45}^{\left(3\right)}-X_{34}^{\left(4\right)})}_{\varphi }& D_{55}^{\left(3\right)}-X_{44}^{\left(4\right)}& D_{56}^{\left(3\right)}\\ -{\left(D_{16}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{26}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{36}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{47}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{46}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{56}^{\left(3\right)}\right)}_{\varphi }& D_{66}^{\left(3\right)}\\ -{\left(X_{18}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{28}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{38}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{48}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{58}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{68}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{78}^{\left(3\right)}\right)}_{\varphi }\\ -{(D_{17}^{\left(3\right)}-D_{13}^{\left(4\right)})}_{\varphi }& -{(D_{27}^{\left(3\right)}-X_{25}^{\left(4\right)})}_{\varphi }& -{\left(D_{37}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{49}^{\left(3\right)}\right)}_{\varphi }& -{(D_{47}^{\left(3\right)}-D_{23}^{\left(4\right)})}_{\varphi }& -{(D_{57}^{\left(3\right)}-X_{45}^{\left(4\right)})}_{\varphi }& -{\left(D_{67}^{\left(3\right)}\right)}_{\varphi }\\ -{(D_{18}^{\left(3\right)}-X_{16}^{\left(4\right)})}_{\varphi }& -{(D_{28}^{\left(3\right)}-X_{26}^{\left(4\right)})}_{\varphi }& -{\left(D_{38}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{4,10}^{\left(3\right)}\right)}_{\varphi }& -{(D_{48}^{\left(3\right)}-X_{36}^{\left(4\right)})}_{\varphi }& -{(D_{58}^{\left(3\right)}-X_{46}^{\left(4\right)})}_{\varphi }& -{\left(D_{68}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(D_{19}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{29}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{39}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{4,11}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{49}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{59}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{69}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{1,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{3,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{4,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{5,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{6,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{7,12}^{\left(3\right)}\right)}_{\varphi }\\ -{(D_{1,10}^{\left(3\right)}-D_{14}^{\left(4\right)})}_{\varphi }& -{(D_{2,10}^{\left(3\right)}-X_{27}^{\left(4\right)})}_{\varphi }& -{\left(D_{3,10}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{4,16}^{\left(2\right)}\right)}_{\varphi }& -{(D_{4,10}^{\left(3\right)}-D_{24}^{\left(4\right)})}_{\varphi }& -{(D_{5,10}^{\left(3\right)}-X_{47}^{\left(4\right)})}_{\varphi }& -{\left(D_{6,10}^{\left(3\right)}\right)}_{\varphi }\\ -{(D_{1,11}^{\left(3\right)}-X_{18}^{\left(4\right)})}_{\varphi }& -{(D_{2,11}^{\left(3\right)}-X_{28}^{\left(4\right)})}_{\varphi }& -{\left(D_{3,11}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{4,17}^{\left(2\right)}\right)}_{\varphi }& -{(D_{4,11}^{\left(3\right)}-X_{38}^{\left(4\right)})}_{\varphi }& -{(D_{5,11}^{\left(3\right)}-X_{48}^{\left(4\right)})}_{\varphi }& -{\left(D_{6,11}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(D_{1,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{2,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{3,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{4,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{4,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{5,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{6,12}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(D_{1,19}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{2,19}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{3,19}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{4,19}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{6,19}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{7,19}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{8,19}^{\left(2\right)}\right)}_{\varphi }\\ -{(D_{1,13}^{\left(3\right)}-D_{15}^{\left(4\right)})}_{\varphi }& -{(D_{2,13}^{\left(3\right)}-X_{29}^{\left(4\right)})}_{\varphi }& -{\left(D_{3,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{4,17}^{\left(3\right)}\right)}_{\varphi }& -{(D_{4,13}^{\left(3\right)}-D_{25}^{\left(4\right)})}_{\varphi }& -{(D_{5,13}^{\left(3\right)}-X_{49}^{\left(4\right)})}_{\varphi }& -{\left(D_{6,13}^{\left(3\right)}\right)}_{\varphi }\\ -{(D_{1,14}^{\left(3\right)}-X_{1,10}^{\left(4\right)})}_{\varphi }& -{(D_{2,14}^{\left(3\right)}-X_{2,10}^{\left(4\right)})}_{\varphi }& -{\left(D_{3,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{4,18}^{\left(3\right)}\right)}_{\varphi }& -{(D_{4,14}^{\left(3\right)}-X_{3,10}^{\left(4\right)})}_{\varphi }& -{(D_{5,14}^{\left(3\right)}-X_{4,10}^{\left(4\right)})}_{\varphi }& -{\left(D_{6,14}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(D_{1,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{2,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{3,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{4,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{4,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{5,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{6,15}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{1,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{3,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{4,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{5,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{6,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{7,20}^{\left(3\right)}\right)}_{\varphi }\end{array}\right.\\ \hfill \begin{array}{ccccccc}{X}_{18}^{\left(3\right)}& D_{17}^{\left(3\right)}-D_{13}^{\left(4\right)}& D_{18}^{\left(3\right)}-X_{16}^{\left(4\right)}& D_{19}^{\left(3\right)}& X_{1,12}^{\left(3\right)}& D_{1,10}^{\left(3\right)}-D_{14}^{\left(4\right)}& D_{1,11}^{\left(3\right)}-X_{18}^{\left(4\right)}\\ X_{28}^{\left(3\right)}& D_{27}^{\left(3\right)}-X_{25}^{\left(4\right)}& D_{28}^{\left(3\right)}-X_{26}^{\left(4\right)}& D_{29}^{\left(3\right)}& X_{2,12}^{\left(3\right)}& D_{2,10}^{\left(3\right)}-X_{27}^{\left(4\right)}& D_{2,11}^{\left(3\right)}-X_{28}^{\left(4\right)}\\ X_{38}^{\left(3\right)}& D_{37}^{\left(3\right)}& D_{38}^{\left(3\right)}& D_{39}^{\left(3\right)}& X_{3,12}^{\left(3\right)}& D_{3,10}^{\left(3\right)}& D_{3,11}^{\left(3\right)}\\ X_{48}^{\left(3\right)}& X_{49}^{\left(3\right)}& X_{4,10}^{\left(3\right)}& X_{4,11}^{\left(3\right)}& X_{4,12}^{\left(3\right)}& D_{4,16}^{\left(2\right)}& D_{4,17}^{\left(2\right)}\\ X_{58}^{\left(3\right)}& D_{47}^{\left(3\right)}-D_{23}^{\left(4\right)}& D_{48}^{\left(3\right)}-X_{36}^{\left(4\right)}& D_{49}^{\left(3\right)}& X_{5,12}^{\left(3\right)}& D_{4,10}^{\left(3\right)}-D_{24}^{\left(4\right)}& D_{4,11}^{\left(3\right)}-X_{38}^{\left(4\right)}\\ X_{68}^{\left(3\right)}& D_{57}^{\left(3\right)}-X_{45}^{\left(4\right)}& D_{58}^{\left(3\right)}-X_{46}^{\left(4\right)}& D_{59}^{\left(3\right)}& X_{6,12}^{\left(3\right)}& D_{5,10}^{\left(3\right)}-X_{47}^{\left(4\right)}& D_{5,11}^{\left(3\right)}-X_{48}^{\left(4\right)}\\ X_{78}^{\left(3\right)}& D_{67}^{\left(3\right)}& D_{68}^{\left(3\right)}& D_{69}^{\left(3\right)}& X_{7,12}^{\left(3\right)}& D_{6,10}^{\left(3\right)}& D_{6,11}^{\left(3\right)}\\ X_{88}^{\left(3\right)}& X_{89}^{\left(3\right)}& X_{8,10}^{\left(3\right)}& X_{8,11}^{\left(3\right)}& X_{8,12}^{\left(3\right)}& D_{9,16}^{\left(2\right)}& D_{9,17}^{\left(2\right)}\\ -{\left(X_{89}^{\left(3\right)}\right)}_{\varphi }& D_{77}^{\left(3\right)}-D_{33}^{\left(4\right)}& D_{78}^{\left(3\right)}-X_{56}^{\left(4\right)}& D_{79}^{\left(3\right)}& X_{9,12}^{\left(3\right)}& D_{7,10}^{\left(3\right)}-D_{34}^{\left(4\right)}& D_{7,11}^{\left(3\right)}-X_{58}^{\left(4\right)}\\ -{\left(X_{8,10}^{\left(3\right)}\right)}_{\varphi }& -{(D_{78}^{\left(3\right)}-X_{56}^{\left(4\right)})}_{\varphi }& D_{88}^{\left(3\right)}-X_{66}^{\left(4\right)}& D_{89}^{\left(3\right)}& X_{10,12}^{\left(3\right)}& D_{8,10}^{\left(3\right)}-X_{67}^{\left(4\right)}& D_{8,11}^{\left(3\right)}-X_{68}^{\left(4\right)}\\ -{\left(X_{8,11}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{79}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{89}^{\left(3\right)}\right)}_{\varphi }& D_{99}^{\left(3\right)}& X_{11,12}^{\left(3\right)}& D_{9,10}^{\left(3\right)}& D_{9,11}^{\left(3\right)}\\ -{\left(X_{8,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{9,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{10,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{11,12}^{\left(3\right)}\right)}_{\varphi }& X_{12,12}^{\left(3\right)}& D_{14,16}^{\left(2\right)}& D_{14,17}^{\left(2\right)}\\ -{\left(D_{9,16}^{\left(2\right)}\right)}_{\varphi }& -{(D_{7,10}^{\left(3\right)}-D_{34}^{\left(4\right)})}_{\varphi }& -{(D_{8,10}^{\left(3\right)}-X_{67}^{\left(4\right)})}_{\varphi }& -{\left(D_{9,10}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{14,16}^{\left(2\right)}\right)}_{\varphi }& D_{10,10}^{\left(3\right)}-D_{44}^{\left(4\right)}& D_{10,11}^{\left(3\right)}-X_{78}^{\left(4\right)}\\ -{\left(D_{9,17}^{\left(2\right)}\right)}_{\varphi }& -{(D_{7,11}^{\left(3\right)}-X_{58}^{\left(4\right)})}_{\varphi }& -{(D_{8,11}^{\left(3\right)}-X_{68}^{\left(4\right)})}_{\varphi }& -{\left(D_{9,11}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{14,17}^{\left(2\right)}\right)}_{\varphi }& -{(D_{10,11}^{\left(3\right)}-X_{78}^{\left(4\right)})}_{\varphi }& D_{11,11}^{\left(3\right)}-X_{88}^{\left(4\right)}\\ -{\left(D_{9,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{7,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{8,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{9,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{14,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{10,12}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{11,12}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(D_{9,19}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{11,19}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{12,19}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{13,19}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{14,19}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{16,19}^{\left(2\right)}\right)}_{\varphi }& -{\left(D_{17,19}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{8,17}^{\left(3\right)}\right)}_{\varphi }& -{(D_{7,13}^{\left(3\right)}-D_{35}^{\left(4\right)})}_{\varphi }& -{(D_{8,13}^{\left(3\right)}-X_{69}^{\left(4\right)})}_{\varphi }& -{\left(D_{9,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{12,17}^{\left(3\right)}\right)}_{\varphi }& -{(D_{10,13}^{\left(3\right)}-D_{45}^{\left(4\right)})}_{\varphi }& -{(D_{11,13}^{\left(3\right)}-X_{89}^{\left(4\right)})}_{\varphi }\\ -{\left(X_{8,18}^{\left(3\right)}\right)}_{\varphi }& -{(D_{7,14}^{\left(3\right)}-X_{5,10}^{\left(4\right)})}_{\varphi }& -{(D_{8,14}^{\left(3\right)}-X_{6,10}^{\left(4\right)})}_{\varphi }& -{\left(D_{9,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{12,18}^{\left(3\right)}\right)}_{\varphi }& -{(D_{10,14}^{\left(3\right)}-X_{7,10}^{\left(4\right)})}_{\varphi }& -{(D_{11,14}^{\left(3\right)}-X_{8,10}^{\left(4\right)})}_{\varphi }\\ -{\left(X_{8,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{7,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{8,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{9,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{12,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{10,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{11,15}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{8,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{9,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{10,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{11,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{12,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{13,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{14,20}^{\left(3\right)}\right)}_{\varphi }\end{array}\\ \hfill \left.\begin{array}{cccccc}{D}_{1,12}^{\left(3\right)}& D_{1,19}^{\left(2\right)}& D_{1,13}^{\left(3\right)}-D_{15}^{\left(4\right)}& D_{1,14}^{\left(3\right)}-X_{1,10}^{\left(4\right)}& D_{1,15}^{\left(3\right)}& X_{1,20}^{\left(3\right)}\\ D_{2,12}^{\left(3\right)}& D_{2,19}^{\left(2\right)}& D_{2,13}^{\left(3\right)}-X_{29}^{\left(4\right)}& D_{2,14}^{\left(3\right)}-X_{2,10}^{\left(4\right)}& D_{2,15}^{\left(3\right)}& X_{2,20}^{\left(3\right)}\\ D_{3,12}^{\left(3\right)}& D_{3,19}^{\left(2\right)}& D_{3,13}^{\left(3\right)}& D_{3,14}^{\left(3\right)}& D_{3,15}^{\left(3\right)}& X_{3,20}^{\left(3\right)}\\ D_{4,18}^{\left(2\right)}& D_{4,19}^{\left(2\right)}& X_{4,17}^{\left(3\right)}& X_{4,18}^{\left(3\right)}& X_{4,19}^{\left(3\right)}& X_{4,20}^{\left(3\right)}\\ D_{4,12}^{\left(3\right)}& D_{6,19}^{\left(2\right)}& D_{4,13}^{\left(3\right)}-D_{25}^{\left(4\right)}& D_{4,14}^{\left(3\right)}-X_{3,10}^{\left(4\right)}& D_{4,15}^{\left(3\right)}& X_{5,20}^{\left(3\right)}\\ D_{5,12}^{\left(3\right)}& D_{7,19}^{\left(2\right)}& D_{5,13}^{\left(3\right)}-X_{49}^{\left(4\right)}& D_{5,14}^{\left(3\right)}-X_{4,10}^{\left(4\right)}& D_{5,15}^{\left(3\right)}& X_{6,20}^{\left(3\right)}\\ D_{6,12}^{\left(3\right)}& D_{8,19}^{\left(2\right)}& D_{6,13}^{\left(3\right)}& D_{6,14}^{\left(3\right)}& D_{6,15}^{\left(3\right)}& X_{7,20}^{\left(3\right)}\\ D_{9,18}^{\left(2\right)}& D_{9,19}^{\left(2\right)}& X_{8,17}^{\left(3\right)}& X_{8,18}^{\left(3\right)}& X_{8,19}^{\left(3\right)}& X_{8,20}^{\left(3\right)}\\ D_{7,12}^{\left(3\right)}& D_{11,19}^{\left(2\right)}& D_{7,13}^{\left(3\right)}-D_{35}^{\left(4\right)}& D_{7,14}^{\left(3\right)}-X_{5,10}^{\left(4\right)}& D_{7,15}^{\left(3\right)}& X_{9,20}^{\left(3\right)}\\ D_{8,12}^{\left(3\right)}& D_{12,19}^{\left(2\right)}& D_{8,13}^{\left(3\right)}-X_{69}^{\left(4\right)}& D_{8,14}^{\left(3\right)}-X_{6,10}^{\left(4\right)}& D_{8,15}^{\left(3\right)}& X_{10,20}^{\left(3\right)}\\ D_{9,12}^{\left(3\right)}& D_{13,19}^{\left(2\right)}& D_{9,13}^{\left(3\right)}& D_{9,14}^{\left(3\right)}& D_{9,15}^{\left(3\right)}& X_{11,20}^{\left(3\right)}\\ D_{14,18}^{\left(2\right)}& D_{14,19}^{\left(2\right)}& X_{12,17}^{\left(3\right)}& X_{12,18}^{\left(3\right)}& X_{12,19}^{\left(3\right)}& X_{12,20}^{\left(3\right)}\\ D_{10,12}^{\left(3\right)}& D_{16,19}^{\left(2\right)}& D_{10,13}^{\left(3\right)}-D_{45}^{\left(4\right)}& D_{10,14}^{\left(3\right)}-X_{7,10}^{\left(4\right)}& D_{10,15}^{\left(3\right)}& X_{13,20}^{\left(3\right)}\\ D_{11,12}^{\left(3\right)}& D_{17,19}^{\left(2\right)}& D_{11,13}^{\left(3\right)}-X_{89}^{\left(4\right)}& D_{11,14}^{\left(3\right)}-X_{8,10}^{\left(4\right)}& D_{11,15}^{\left(3\right)}& X_{14,20}^{\left(3\right)}\\ D_{12,12}^{\left(3\right)}& D_{18,19}^{\left(2\right)}& D_{12,13}^{\left(3\right)}& D_{12,14}^{\left(3\right)}& D_{12,15}^{\left(3\right)}& X_{15,20}^{\left(3\right)}\\ -{\left(D_{18,19}^{\left(2\right)}\right)}_{\varphi }& D_{19,19}^{\left(2\right)}& X_{16,17}^{\left(3\right)}& X_{16,18}^{\left(3\right)}& X_{16,19}^{\left(3\right)}& X_{16,20}^{\left(3\right)}\\ -{\left(D_{12,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{16,17}^{\left(3\right)}\right)}_{\varphi }& D_{13,13}^{\left(3\right)}-D_{55}^{\left(4\right)}& D_{13,14}^{\left(3\right)}-X_{9,10}^{\left(4\right)}& D_{13,15}^{\left(3\right)}& X_{17,20}^{\left(3\right)}\\ -{\left(D_{12,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{16,18}^{\left(3\right)}\right)}_{\varphi }& -{(D_{13,14}^{\left(3\right)}-X_{9,10}^{\left(4\right)})}_{\varphi }& D_{14,14}^{\left(3\right)}-X_{10,10}^{\left(4\right)}& D_{14,15}^{\left(3\right)}& X_{18,20}^{\left(3\right)}\\ -{\left(D_{12,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{16,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{13,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{14,15}^{\left(3\right)}\right)}_{\varphi }& D_{15,15}^{\left(3\right)}& X_{19,20}^{\left(3\right)}\\ -{\left(X_{15,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{16,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{17,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{18,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{19,20}^{\left(3\right)}\right)}_{\varphi }& X_{20,20}^{\left(3\right)}\end{array}\right),\end{array}$$

(9)

$$\begin{array}{c}\hfill {\widehat{X}}_4=\left(\begin{array}{ccccccc}{D}_{11}^{\left(4\right)}& X_{12}^{\left(4\right)}& D_{12}^{\left(4\right)}& X_{14}^{\left(4\right)}& D_{13}^{\left(4\right)}& X_{16}^{\left(4\right)}& D_{14}^{\left(4\right)}\\ -{\left(X_{12}^{\left(4\right)}\right)}_{\varphi }& X_{22}^{\left(4\right)}& X_{23}^{\left(4\right)}& X_{24}^{\left(4\right)}& X_{25}^{\left(4\right)}& X_{26}^{\left(4\right)}& X_{27}^{\left(4\right)}\\ -{\left(D_{12}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{23}^{\left(4\right)}\right)}_{\varphi }& D_{22}^{\left(4\right)}& X_{34}^{\left(4\right)}& D_{23}^{\left(4\right)}& X_{36}^{\left(4\right)}& D_{24}^{\left(4\right)}\\ -{\left(X_{14}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{24}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{34}^{\left(4\right)}\right)}_{\varphi }& X_{44}^{\left(4\right)}& X_{45}^{\left(4\right)}& X_{46}^{\left(4\right)}& X_{47}^{\left(4\right)}\\ -{\left(D_{13}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{25}^{\left(4\right)}\right)}_{\varphi }& -{\left(D_{23}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{45}^{\left(4\right)}\right)}_{\varphi }& D_{33}^{\left(4\right)}& X_{56}^{\left(4\right)}& D_{34}^{\left(4\right)}\\ -{\left(X_{16}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{26}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{36}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{46}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{56}^{\left(4\right)}\right)}_{\varphi }& X_{66}^{\left(4\right)}& X_{67}^{\left(4\right)}\\ -{\left(D_{14}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{27}^{\left(4\right)}\right)}_{\varphi }& -{\left(D_{24}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{47}^{\left(4\right)}\right)}_{\varphi }& -{\left(D_{34}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{67}^{\left(4\right)}\right)}_{\varphi }& D_{44}^{\left(4\right)}\\ -{\left(X_{18}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{28}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{38}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{48}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{58}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{68}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{78}^{\left(4\right)}\right)}_{\varphi }\\ -{\left(D_{15}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{29}^{\left(4\right)}\right)}_{\varphi }& -{\left(D_{25}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{49}^{\left(4\right)}\right)}_{\varphi }& -{\left(D_{35}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{69}^{\left(4\right)}\right)}_{\varphi }& -{\left(D_{45}^{\left(4\right)}\right)}_{\varphi }\\ -{\left(X_{1,10}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{2,10}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{3,10}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{4,10}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{5,10}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{6,10}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{7,10}^{\left(4\right)}\right)}_{\varphi }\\ -{\left(D_{16}^{\left(4\right)}\right)}_{\varphi }& -{\left(D_{2,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{26}^{\left(4\right)}\right)}_{\varphi }& -{\left(D_{5,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{36}^{\left(4\right)}\right)}_{\varphi }& -{\left(D_{8,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{46}^{\left(4\right)}\right)}_{\varphi }\\ -{\left(D_{1,17}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{2,17}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{4,17}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{5,17}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{7,17}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{8,17}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{10,17}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(D_{17}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{2,13}^{\left(4\right)}\right)}_{\varphi }& -{\left(D_{27}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{4,13}^{\left(4\right)}\right)}_{\varphi }& -{\left(D_{37}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{6,13}^{\left(4\right)}\right)}_{\varphi }& -{\left(D_{47}^{\left(4\right)}\right)}_{\varphi }\\ -{\left(X_{1,14}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{2,14}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{3,14}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{4,14}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{5,14}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{6,14}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{7,14}^{\left(4\right)}\right)}_{\varphi }\end{array}\right.\\ \hfill \left.\begin{array}{ccccccc}{X}_{18}^{\left(4\right)}& D_{15}^{\left(4\right)}& X_{1,10}^{\left(4\right)}& D_{16}^{\left(4\right)}& D_{1,17}^{\left(3\right)}& D_{17}^{\left(4\right)}& X_{1,14}^{\left(4\right)}\\ X_{28}^{\left(4\right)}& X_{29}^{\left(4\right)}& X_{2,10}^{\left(4\right)}& D_{2,16}^{\left(3\right)}& D_{2,17}^{\left(3\right)}& X_{2,13}^{\left(4\right)}& X_{2,14}^{\left(4\right)}\\ X_{38}^{\left(4\right)}& D_{25}^{\left(4\right)}& X_{3,10}^{\left(4\right)}& D_{26}^{\left(4\right)}& D_{4,17}^{\left(3\right)}& D_{27}^{\left(4\right)}& X_{3,14}^{\left(4\right)}\\ X_{48}^{\left(4\right)}& X_{49}^{\left(4\right)}& X_{4,10}^{\left(4\right)}& D_{5,16}^{\left(3\right)}& D_{5,17}^{\left(3\right)}& X_{4,13}^{\left(4\right)}& X_{4,14}^{\left(4\right)}\\ X_{58}^{\left(4\right)}& D_{35}^{\left(4\right)}& X_{5,10}^{\left(4\right)}& D_{36}^{\left(4\right)}& D_{7,17}^{\left(3\right)}& D_{37}^{\left(4\right)}& X_{5,14}^{\left(4\right)}\\ X_{68}^{\left(4\right)}& X_{69}^{\left(4\right)}& X_{6,10}^{\left(4\right)}& D_{8,16}^{\left(3\right)}& D_{8,17}^{\left(3\right)}& X_{6,13}^{\left(4\right)}& X_{6,14}^{\left(4\right)}\\ X_{78}^{\left(4\right)}& D_{45}^{\left(4\right)}& X_{7,10}^{\left(4\right)}& D_{46}^{\left(4\right)}& D_{10,17}^{\left(3\right)}& D_{47}^{\left(4\right)}& X_{7,14}^{\left(4\right)}\\ X_{88}^{\left(4\right)}& X_{89}^{\left(4\right)}& X_{8,10}^{\left(4\right)}& D_{11,16}^{\left(3\right)}& D_{11,17}^{\left(3\right)}& X_{8,13}^{\left(4\right)}& X_{8,14}^{\left(4\right)}\\ -{\left(X_{89}^{\left(4\right)}\right)}_{\varphi }& D_{55}^{\left(4\right)}& X_{9,10}^{\left(4\right)}& D_{56}^{\left(4\right)}& D_{13,17}^{\left(3\right)}& D_{57}^{\left(4\right)}& X_{9,14}^{\left(4\right)}\\ -{\left(X_{8,10}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{9,10}^{\left(4\right)}\right)}_{\varphi }& X_{10,10}^{\left(4\right)}& D_{14,16}^{\left(3\right)}& D_{14,17}^{\left(3\right)}& X_{10,13}^{\left(4\right)}& X_{10,14}^{\left(4\right)}\\ -{\left(D_{11,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{56}^{\left(4\right)}\right)}_{\varphi }& -{\left(D_{14,16}^{\left(3\right)}\right)}_{\varphi }& D_{66}^{\left(4\right)}& D_{16,17}^{\left(3\right)}& D_{67}^{\left(4\right)}& X_{11,14}^{\left(4\right)}\\ -{\left(D_{11,17}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{13,17}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{14,17}^{\left(3\right)}\right)}_{\varphi }& -{\left(D_{16,17}^{\left(3\right)}\right)}_{\varphi }& D_{17,17}^{\left(3\right)}& X_{12,13}^{\left(4\right)}& X_{12,14}^{\left(4\right)}\\ -{\left(X_{8,13}^{\left(4\right)}\right)}_{\varphi }& -{\left(D_{57}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{10,13}^{\left(4\right)}\right)}_{\varphi }& -{\left(D_{67}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{12,13}^{\left(4\right)}\right)}_{\varphi }& D_{77}^{\left(4\right)}& X_{13,14}^{\left(4\right)}\\ -{\left(X_{8,14}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{9,14}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{10,14}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{11,14}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{12,14}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{13,14}^{\left(4\right)}\right)}_{\varphi }& X_{14,14}^{\left(4\right)}\end{array}\right),\end{array}$$

(10)

where $D_{kj}^{\left(i\right)}={\widehat{P}}_i{C}_i{\left({\widehat{P}}_i\right)}_{\varphi },(i=1,2,3,4)$ are defined in [20], and the remaining $X_{l_1{m}_1}^{\left(1\right)},X_{l_2{m}_2}^{\left(2\right)},X_{l_3{m}_3}^{\left(3\right)},X_{l_4{m}_4}^{\left(4\right)}$ are arbitrary matrices over $\mathbb{H}$ with appropriate sizes.

Proof. 

According to the idea of [20], we assume ${\widehat{X}}_1$, ${\widehat{X}}_2$, ${\widehat{X}}_3$, ${\widehat{X}}_4$ have the following form:

$$\begin{array}{c}\hfill {\widehat{X}}_1=-{\left({\widehat{X}}_1\right)}_{\varphi }=\left(\begin{array}{cccc}{X}_{11}^{\left(1\right)}& X_{12}^{\left(1\right)}& \cdots & X_{18}^{\left(1\right)}\\ -{\left(X_{12}^{\left(1\right)}\right)}_{\varphi }& X_{22}^{\left(1\right)}& \cdots & X_{28}^{\left(1\right)}\\ ⋮& ⋮& \cdots & ⋮\\ -{\left(X_{18}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{28}^{\left(1\right)}\right)}_{\varphi }& \cdots & X_{88}^{\left(1\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\widehat{X}}_2=-{\left({\widehat{X}}_2\right)}_{\varphi }=\left(\begin{array}{cccc}{X}_{11}^{\left(2\right)}& X_{12}^{\left(2\right)}& \cdots & X_{1,18}^{\left(2\right)}\\ -{\left(X_{12}^{\left(2\right)}\right)}_{\varphi }& X_{22}^{\left(2\right)}& \cdots & X_{2,18}^{\left(2\right)}\\ ⋮& ⋮& \cdots & ⋮\\ -{\left(X_{1,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{2,18}^{\left(2\right)}\right)}_{\varphi }& \cdots & X_{18,18}^{\left(2\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\widehat{X}}_3=-{\left({\widehat{X}}_3\right)}_{\varphi }=\left(\begin{array}{cccc}{X}_{11}^{\left(3\right)}& X_{12}^{\left(3\right)}& \cdots & X_{1,20}^{\left(3\right)}\\ -{\left(X_{12}^{\left(3\right)}\right)}_{\varphi }& X_{22}^{\left(3\right)}& \cdots & X_{2,20}^{\left(3\right)}\\ ⋮& ⋮& \cdots & ⋮\\ -{\left(X_{1,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,20}^{\left(3\right)}\right)}_{\varphi }& \cdots & X_{20,20}^{\left(3\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\widehat{X}}_4=-{\left({\widehat{X}}_4\right)}_{\varphi }=\left(\begin{array}{cccc}{X}_{11}^{\left(4\right)}& X_{12}^{\left(4\right)}& \cdots & X_{1,14}^{\left(4\right)}\\ -{\left(X_{12}^{\left(4\right)}\right)}_{\varphi }& X_{22}^{\left(4\right)}& \cdots & X_{2,14}^{\left(4\right)}\\ ⋮& ⋮& \cdots & ⋮\\ -{\left(X_{1,14}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{2,14}^{\left(4\right)}\right)}_{\varphi }& \cdots & X_{14,14}^{\left(4\right)}\end{array}\right).\end{array}$$

Putting ${\widehat{X}}_1$, ${\widehat{X}}_2$, ${\widehat{X}}_3$, ${\widehat{X}}_4$ into the Equation (4) yields

$$\begin{array}{c}\hfill {\left(D_{ij}^{\left(1\right)}\right)}_{14\times 14}=(D_1^{\left(1\right)},D_2^{\left(1\right)}),\end{array}$$

(11)

$$\begin{array}{c}\hfill {\left(D_{ij}^{\left(2\right)}\right)}_{20\times 20}=(D_1^{\left(2\right)},D_2^{\left(2\right)},D_3^{\left(2\right)}),\end{array}$$

(12)

$$\begin{array}{c}\hfill {\left(D_{ij}^{\left(3\right)}\right)}_{18\times 18}=(D_1^{\left(3\right)},D_2^{\left(3\right)},D_3^{\left(3\right)}),\end{array}$$

(13)

$$\begin{array}{c}\hfill {\left(D_{ij}^{\left(4\right)}\right)}_{8\times 8}=\left(\begin{array}{cccccccc}{X}_{11}^{\left(4\right)}& X_{13}^{\left(4\right)}& X_{15}^{\left(4\right)}& X_{17}^{\left(4\right)}& X_{19}^{\left(4\right)}& X_{1,11}^{\left(4\right)}& X_{1,13}^{\left(4\right)}& 0\\ -{\left(X_{13}^{\left(4\right)}\right)}_{\varphi }& X_{33}^{\left(4\right)}& X_{35}^{\left(4\right)}& X_{37}^{\left(4\right)}& X_{39}^{\left(4\right)}& X_{3,11}^{\left(4\right)}& X_{3,13}^{\left(4\right)}& 0\\ -{\left(X_{15}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{35}^{\left(4\right)}\right)}_{\varphi }& X_{55}^{\left(4\right)}& X_{57}^{\left(4\right)}& X_{59}^{\left(4\right)}& X_{5,11}^{\left(4\right)}& X_{5,13}^{\left(4\right)}& 0\\ -{\left(X_{17}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{37}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{57}^{\left(4\right)}\right)}_{\varphi }& X_{77}^{\left(4\right)}& X_{79}^{\left(4\right)}& X_{7,11}^{\left(4\right)}& X_{7,13}^{\left(4\right)}& 0\\ -{\left(X_{19}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{39}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{59}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{79}^{\left(4\right)}\right)}_{\varphi }& X_{99}^{\left(4\right)}& X_{9,11}^{\left(4\right)}& X_{9,13}^{\left(4\right)}& 0\\ -{\left(X_{1,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{3,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{5,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{7,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{9,11}^{\left(4\right)}\right)}_{\varphi }& X_{11,11}^{\left(4\right)}& X_{11,13}^{\left(4\right)}& 0\\ -{\left(X_{1,13}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{3,13}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{5,13}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{7,13}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{9,13}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{11,13}^{\left(4\right)}\right)}_{\varphi }& X_{13,13}^{\left(4\right)}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

(14)

where

$$\begin{array}{c}\hfill D_1^{\left(1\right)}=\left(\begin{array}{ccccc}{X}_{11}^{\left(1\right)}+X_{11}^{\left(2\right)}& X_{12}^{\left(1\right)}+X_{12}^{\left(2\right)}& X_{13}^{\left(1\right)}+X_{13}^{\left(2\right)}& X_{14}^{\left(1\right)}+X_{14}^{\left(2\right)}& X_{15}^{\left(1\right)}+X_{15}^{\left(2\right)}\\ -{(X_{12}^{\left(1\right)}+X_{12}^{\left(2\right)})}_{\varphi }& X_{22}^{\left(1\right)}+X_{22}^{\left(2\right)}& X_{23}^{\left(1\right)}+X_{23}^{\left(2\right)}& X_{24}^{\left(1\right)}+X_{24}^{\left(2\right)}& X_{25}^{\left(1\right)}+X_{25}^{\left(2\right)}\\ -{(X_{13}^{\left(1\right)}+X_{13}^{\left(2\right)})}_{\varphi }& -{(X_{23}^{\left(1\right)}+X_{23}^{\left(2\right)})}_{\varphi }& X_{33}^{\left(1\right)}+X_{33}^{\left(2\right)}& X_{34}^{\left(1\right)}+X_{34}^{\left(2\right)}& X_{35}^{\left(1\right)}+X_{35}^{\left(2\right)}\\ -{(X_{14}^{\left(1\right)}+X_{14}^{\left(2\right)})}_{\varphi }& -{(X_{24}^{\left(1\right)}+X_{24}^{\left(2\right)})}_{\varphi }& -{(X_{34}^{\left(1\right)}+X_{34}^{\left(2\right)})}_{\varphi }& X_{44}^{\left(1\right)}+X_{44}^{\left(2\right)}& X_{45}^{\left(1\right)}+X_{45}^{\left(2\right)}\\ -{(X_{15}^{\left(1\right)}+X_{15}^{\left(2\right)})}_{\varphi }& -{(X_{25}^{\left(1\right)}+X_{25}^{\left(2\right)})}_{\varphi }& -{(X_{35}^{\left(1\right)}+X_{35}^{\left(2\right)})}_{\varphi }& -{(X_{45}^{\left(1\right)}+X_{45}^{\left(2\right)})}_{\varphi }& X_{55}^{\left(1\right)}+X_{55}^{\left(2\right)}\\ -{(X_{16}^{\left(1\right)}+X_{16}^{\left(2\right)})}_{\varphi }& -{(X_{26}^{\left(1\right)}+X_{26}^{\left(2\right)})}_{\varphi }& -{(X_{36}^{\left(1\right)}+X_{36}^{\left(2\right)})}_{\varphi }& -{(X_{46}^{\left(1\right)}+X_{46}^{\left(2\right)})}_{\varphi }& -{(X_{56}^{\left(1\right)}+X_{56}^{\left(2\right)})}_{\varphi }\\ -{\left(X_{17}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{27}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{37}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{47}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{57}^{\left(1\right)}\right)}_{\varphi }\\ -{\left(X_{17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{27}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{37}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{47}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{57}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{28}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{38}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{48}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{58}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{19}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{29}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{39}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{49}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{59}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{1,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{2,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{3,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{4,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{5,10}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{1,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{2,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{3,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{4,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{5,11}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{1,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{2,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{3,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{4,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{5,12}^{\left(2\right)}\right)}_{\varphi }\\ 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill D_2^{\left(1\right)}=\left(\begin{array}{ccccccccc}{X}_{16}^{\left(1\right)}+X_{16}^{\left(2\right)}& X_{17}^{\left(1\right)}& X_{17}^{\left(2\right)}& X_{18}^{\left(2\right)}& X_{19}^{\left(2\right)}& X_{1,10}^{\left(2\right)}& X_{1,11}^{\left(2\right)}& X_{1,12}^{\left(2\right)}& 0\\ X_{26}^{\left(1\right)}+X_{26}^{\left(2\right)}& X_{27}^{\left(1\right)}& X_{27}^{\left(2\right)}& X_{28}^{\left(2\right)}& X_{29}^{\left(2\right)}& X_{2,10}^{\left(2\right)}& X_{2,11}^{\left(2\right)}& X_{2,12}^{\left(2\right)}& 0\\ X_{36}^{\left(1\right)}+X_{36}^{\left(2\right)}& X_{37}^{\left(1\right)}& X_{37}^{\left(2\right)}& X_{38}^{\left(2\right)}& X_{39}^{\left(2\right)}& X_{3,10}^{\left(2\right)}& X_{3,11}^{\left(2\right)}& X_{3,12}^{\left(2\right)}& 0\\ X_{46}^{\left(1\right)}+X_{46}^{\left(2\right)}& X_{47}^{\left(1\right)}& X_{47}^{\left(2\right)}& X_{48}^{\left(2\right)}& X_{49}^{\left(2\right)}& X_{4,10}^{\left(2\right)}& X_{4,11}^{\left(2\right)}& X_{4,12}^{\left(2\right)}& 0\\ X_{56}^{\left(1\right)}+X_{56}^{\left(2\right)}& X_{57}^{\left(1\right)}& X_{57}^{\left(2\right)}& X_{58}^{\left(2\right)}& X_{59}^{\left(2\right)}& X_{5,10}^{\left(2\right)}& X_{5,11}^{\left(2\right)}& X_{5,12}^{\left(2\right)}& 0\\ X_{66}^{\left(1\right)}+X_{66}^{\left(2\right)}& X_{67}^{\left(1\right)}& X_{67}^{\left(2\right)}& X_{68}^{\left(2\right)}& X_{69}^{\left(2\right)}& X_{6,10}^{\left(2\right)}& X_{6,11}^{\left(2\right)}& X_{6,12}^{\left(2\right)}& 0\\ -{\left(X_{67}^{\left(1\right)}\right)}_{\varphi }& X_{77}^{\left(1\right)}& 0& 0& 0& 0& 0& 0& 0\\ -{\left(X_{67}^{\left(2\right)}\right)}_{\varphi }& 0& X_{77}^{\left(2\right)}& X_{78}^{\left(2\right)}& X_{79}^{\left(2\right)}& X_{7,10}^{\left(2\right)}& X_{7,11}^{\left(2\right)}& X_{7,12}^{\left(2\right)}& 0\\ -{\left(X_{68}^{\left(2\right)}\right)}_{\varphi }& 0& -{\left(X_{78}^{\left(2\right)}\right)}_{\varphi }& X_{88}^{\left(2\right)}& X_{89}^{\left(2\right)}& X_{8,10}^{\left(2\right)}& X_{8,11}^{\left(2\right)}& X_{8,12}^{\left(2\right)}& 0\\ -{\left(X_{69}^{\left(2\right)}\right)}_{\varphi }& 0& -{\left(X_{79}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{89}^{\left(2\right)}\right)}_{\varphi }& X_{9,9}^{\left(2\right)}& X_{9,10}^{\left(2\right)}& X_{9,11}^{\left(2\right)}& X_{9,12}^{\left(2\right)}& 0\\ -{\left(X_{6,10}^{\left(2\right)}\right)}_{\varphi }& 0& -{\left(X_{7,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{8,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{9,10}^{\left(2\right)}\right)}_{\varphi }& X_{10,10}^{\left(2\right)}& X_{10,11}^{\left(2\right)}& X_{10,12}^{\left(2\right)}& 0\\ -{\left(X_{6,11}^{\left(2\right)}\right)}_{\varphi }& 0& -{\left(X_{7,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{8,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{9,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{10,11}^{\left(2\right)}\right)}_{\varphi }& X_{11,11}^{\left(2\right)}& X_{11,12}^{\left(2\right)}& 0\\ -{\left(X_{6,12}^{\left(2\right)}\right)}_{\varphi }& 0& -{\left(X_{7,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{8,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{9,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{10,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{11,12}^{\left(2\right)}\right)}_{\varphi }& X_{12,12}^{\left(2\right)}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill D_1^{\left(2\right)}=\left(\begin{array}{cccccc}{X}_{11}^{\left(2\right)}+X_{11}^{\left(3\right)}& X_{12}^{\left(2\right)}+X_{12}^{\left(3\right)}& X_{13}^{\left(2\right)}+X_{13}^{\left(3\right)}& X_{14}^{\left(2\right)}+X_{14}^{\left(3\right)}& X_{15}^{\left(2\right)}& X_{17}^{\left(2\right)}+X_{15}^{\left(3\right)}\\ -{(X_{12}^{\left(2\right)}+X_{12}^{\left(3\right)})}_{\varphi }& X_{22}^{\left(2\right)}+X_{22}^{\left(3\right)}& X_{23}^{\left(2\right)}+X_{23}^{\left(3\right)}& X_{24}^{\left(2\right)}+X_{24}^{\left(3\right)}& X_{25}^{\left(2\right)}& X_{27}^{\left(2\right)}+X_{25}^{\left(3\right)}\\ -{(X_{13}^{\left(2\right)}+X_{13}^{\left(3\right)})}_{\varphi }& -{(X_{23}^{\left(2\right)}+X_{23}^{\left(3\right)})}_{\varphi }& X_{33}^{\left(2\right)}+X_{33}^{\left(3\right)}& X_{34}^{\left(2\right)}+X_{34}^{\left(3\right)}& X_{35}^{\left(2\right)}& X_{37}^{\left(2\right)}+X_{35}^{\left(3\right)}\\ -{(X_{14}^{\left(2\right)}+X_{14}^{\left(3\right)})}_{\varphi }& -{(X_{24}^{\left(2\right)}+X_{24}^{\left(3\right)})}_{\varphi }& -{(X_{34}^{\left(2\right)}+X_{34}^{\left(3\right)})}_{\varphi }& X_{44}^{\left(2\right)}+X_{44}^{\left(3\right)}& X_{45}^{\left(2\right)}& X_{47}^{\left(2\right)}+X_{45}^{\left(3\right)}\\ -{\left(X_{15}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{25}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{35}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{45}^{\left(2\right)}\right)}_{\varphi }& X_{55}^{\left(2\right)}& X_{57}^{\left(2\right)}\\ -{(X_{17}^{\left(2\right)}+X_{15}^{\left(3\right)})}_{\varphi }& -{(X_{27}^{\left(2\right)}+X_{25}^{\left(3\right)})}_{\varphi }& -{(X_{37}^{\left(2\right)}+X_{35}^{\left(3\right)})}_{\varphi }& -{(X_{47}^{\left(2\right)}+X_{45}^{\left(3\right)})}_{\varphi }& -{\left(X_{57}^{\left(2\right)}\right)}_{\varphi }& X_{77}^{\left(2\right)}+X_{55}^{\left(3\right)}\\ -{(X_{18}^{\left(2\right)}+X_{16}^{\left(3\right)})}_{\varphi }& -{(X_{28}^{\left(2\right)}+X_{26}^{\left(3\right)})}_{\varphi }& -{(X_{38}^{\left(2\right)}+X_{36}^{\left(3\right)})}_{\varphi }& -{(X_{48}^{\left(2\right)}+X_{46}^{\left(3\right)})}_{\varphi }& -{\left(X_{58}^{\left(2\right)}\right)}_{\varphi }& -{(X_{78}^{\left(2\right)}+X_{56}^{\left(3\right)})}_{\varphi }\\ -{(X_{19}^{\left(2\right)}+X_{17}^{\left(3\right)})}_{\varphi }& -{(X_{29}^{\left(2\right)}+X_{27}^{\left(3\right)})}_{\varphi }& -{(X_{39}^{\left(2\right)}+X_{37}^{\left(3\right)})}_{\varphi }& -{(X_{49}^{\left(2\right)}+X_{47}^{\left(3\right)})}_{\varphi }& -{\left(X_{59}^{\left(2\right)}\right)}_{\varphi }& -{(X_{79}^{\left(2\right)}+X_{57}^{\left(3\right)})}_{\varphi }\\ -{(X_{1,10}^{\left(2\right)}+X_{18}^{\left(3\right)})}_{\varphi }& -{(X_{2,10}^{\left(2\right)}+X_{28}^{\left(3\right)})}_{\varphi }& -{(X_{3,10}^{\left(2\right)}+X_{38}^{\left(3\right)})}_{\varphi }& -{(X_{4,10}^{\left(2\right)}+X_{48}^{\left(3\right)})}_{\varphi }& -{\left(X_{5,10}^{\left(2\right)}\right)}_{\varphi }& -{(X_{7,10}^{\left(2\right)}+X_{58}^{\left(3\right)})}_{\varphi }\\ -{\left(X_{1,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{2,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{3,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{4,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{5,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{7,11}^{\left(2\right)}\right)}_{\varphi }\\ -{(X_{1,13}^{\left(2\right)}+X_{19}^{\left(3\right)})}_{\varphi }& -{(X_{2,13}^{\left(2\right)}+X_{29}^{\left(3\right)})}_{\varphi }& -{(X_{3,13}^{\left(2\right)}+X_{39}^{\left(3\right)})}_{\varphi }& -{(X_{4,13}^{\left(2\right)}+X_{49}^{\left(3\right)})}_{\varphi }& -{\left(X_{5,13}^{\left(2\right)}\right)}_{\varphi }& -{(X_{7,13}^{\left(2\right)}+X_{59}^{\left(3\right)})}_{\varphi }\\ -{(X_{1,14}^{\left(2\right)}+X_{1,10}^{\left(3\right)})}_{\varphi }& -{(X_{2,14}^{\left(2\right)}+X_{2,10}^{\left(3\right)})}_{\varphi }& -{(X_{3,14}^{\left(2\right)}+X_{3,10}^{\left(3\right)})}_{\varphi }& -{(X_{4,14}^{\left(2\right)}+X_{4,10}^{\left(3\right)})}_{\varphi }& -{\left(X_{5,14}^{\left(2\right)}\right)}_{\varphi }& -{(X_{7,14}^{\left(2\right)}+X_{5,10}^{\left(3\right)})}_{\varphi }\\ -{(X_{1,15}^{\left(2\right)}+X_{1,11}^{\left(3\right)})}_{\varphi }& -{(X_{2,15}^{\left(2\right)}+X_{2,11}^{\left(3\right)})}_{\varphi }& -{(X_{3,15}^{\left(2\right)}+X_{3,11}^{\left(3\right)})}_{\varphi }& -{(X_{4,15}^{\left(2\right)}+X_{4,11}^{\left(3\right)})}_{\varphi }& -{\left(X_{5,15}^{\left(2\right)}\right)}_{\varphi }& -{(X_{7,15}^{\left(2\right)}+X_{5,11}^{\left(3\right)})}_{\varphi }\\ -{(X_{1,16}^{\left(2\right)}+X_{1,12}^{\left(3\right)})}_{\varphi }& -{(X_{2,16}^{\left(2\right)}+X_{2,12}^{\left(3\right)})}_{\varphi }& -{(X_{3,16}^{\left(2\right)}+X_{3,12}^{\left(3\right)})}_{\varphi }& -{(X_{4,16}^{\left(2\right)}+X_{4,12}^{\left(3\right)})}_{\varphi }& -{\left(X_{5,16}^{\left(2\right)}\right)}_{\varphi }& -{(X_{7,16}^{\left(2\right)}+X_{5,12}^{\left(3\right)})}_{\varphi }\\ -{\left(X_{1,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{2,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{3,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{4,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{5,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{7,17}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{1,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{3,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{4,13}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{5,13}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{1,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{3,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{4,14}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{5,14}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{1,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{3,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{4,15}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{5,15}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{1,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{3,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{4,16}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{5,16}^{\left(3\right)}\right)}_{\varphi }\\ 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill D_2^{\left(2\right)}=\left(\begin{array}{cccccc}{X}_{18}^{\left(2\right)}+X_{16}^{\left(3\right)}& X_{19}^{\left(2\right)}+X_{17}^{\left(3\right)}& X_{1,10}^{\left(2\right)}+X_{18}^{\left(3\right)}& X_{1,11}^{\left(2\right)}& X_{1,13}^{\left(2\right)}+X_{19}^{\left(3\right)}& X_{1,14}^{\left(2\right)}+X_{1,10}^{\left(3\right)}\\ X_{28}^{\left(2\right)}+X_{26}^{\left(3\right)}& X_{29}^{\left(2\right)}+X_{27}^{\left(3\right)}& X_{2,10}^{\left(2\right)}+X_{28}^{\left(3\right)}& X_{2,11}^{\left(2\right)}& X_{2,13}^{\left(2\right)}+X_{29}^{\left(3\right)}& X_{2,14}^{\left(2\right)}+X_{2,10}^{\left(3\right)}\\ X_{38}^{\left(2\right)}+X_{36}^{\left(3\right)}& X_{39}^{\left(2\right)}+X_{37}^{\left(3\right)}& X_{3,10}^{\left(2\right)}+X_{38}^{\left(3\right)}& X_{3,11}^{\left(2\right)}& X_{3,13}^{\left(2\right)}+X_{39}^{\left(3\right)}& X_{3,14}^{\left(2\right)}+X_{3,10}^{\left(3\right)}\\ X_{48}^{\left(2\right)}+X_{46}^{\left(3\right)}& X_{49}^{\left(2\right)}+X_{47}^{\left(3\right)}& X_{4,10}^{\left(2\right)}+X_{48}^{\left(3\right)}& X_{4,11}^{\left(2\right)}& X_{4,13}^{\left(2\right)}+X_{49}^{\left(3\right)}& X_{4,14}^{\left(2\right)}+X_{4,10}^{\left(3\right)}\\ X_{58}^{\left(2\right)}& X_{59}^{\left(2\right)}& X_{5,10}^{\left(2\right)}& X_{5,11}^{\left(2\right)}& X_{5,13}^{\left(2\right)}& X_{5,14}^{\left(2\right)}\\ X_{78}^{\left(2\right)}+X_{56}^{\left(3\right)}& X_{79}^{\left(2\right)}+X_{57}^{\left(3\right)}& X_{7,10}^{\left(2\right)}+X_{58}^{\left(3\right)}& X_{7,11}^{\left(2\right)}& X_{7,13}^{\left(2\right)}+X_{59}^{\left(3\right)}& X_{7,14}^{\left(2\right)}+X_{5,10}^{\left(3\right)}\\ X_{88}^{\left(2\right)}+X_{66}^{\left(3\right)}& X_{89}^{\left(2\right)}+X_{67}^{\left(3\right)}& X_{8,10}^{\left(2\right)}+X_{68}^{\left(3\right)}& X_{8,11}^{\left(2\right)}& X_{8,13}^{\left(2\right)}+X_{69}^{\left(3\right)}& X_{8,14}^{\left(2\right)}+X_{6,10}^{\left(3\right)}\\ -{(X_{89}^{\left(2\right)}+X_{67}^{\left(3\right)})}_{\varphi }& X_{99}^{\left(2\right)}+X_{77}^{\left(3\right)}& X_{9,10}^{\left(2\right)}+X_{78}^{\left(3\right)}& X_{9,11}^{\left(2\right)}& X_{9,13}^{\left(2\right)}+X_{79}^{\left(3\right)}& X_{9,14}^{\left(2\right)}+X_{7,10}^{\left(3\right)}\\ -{(X_{8,10}^{\left(2\right)}+X_{68}^{\left(3\right)})}_{\varphi }& -{(X_{9,10}^{\left(2\right)}+X_{78}^{\left(3\right)})}_{\varphi }& X_{10,10}^{\left(2\right)}+X_{88}^{\left(3\right)}& X_{10,11}^{\left(2\right)}& X_{10,13}^{\left(2\right)}+X_{89}^{\left(3\right)}& X_{10,14}^{\left(2\right)}+X_{8,10}^{\left(3\right)}\\ -{\left(X_{8,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{9,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{10,11}^{\left(2\right)}\right)}_{\varphi }& X_{11,11}^{\left(2\right)}& X_{11,13}^{\left(2\right)}& X_{11,14}^{\left(2\right)}\\ -{(X_{8,13}^{\left(2\right)}+X_{69}^{\left(3\right)})}_{\varphi }& -{(X_{9,13}^{\left(2\right)}+X_{79}^{\left(3\right)})}_{\varphi }& -{(X_{10,13}^{\left(2\right)}+X_{89}^{\left(3\right)})}_{\varphi }& -{\left(X_{11,13}^{\left(2\right)}\right)}_{\varphi }& X_{13,13}^{\left(2\right)}+X_{99}^{\left(3\right)}& X_{13,14}^{\left(2\right)}+X_{9,10}^{\left(3\right)}\\ -{(X_{8,14}^{\left(2\right)}+X_{6,10}^{\left(3\right)})}_{\varphi }& -{(X_{9,14}^{\left(2\right)}+X_{7,10}^{\left(3\right)})}_{\varphi }& -{(X_{10,14}^{\left(2\right)}+X_{8,10}^{\left(3\right)})}_{\varphi }& -{\left(X_{11,14}^{\left(2\right)}\right)}_{\varphi }& -{(X_{13,14}^{\left(2\right)}+X_{9,10}^{\left(3\right)})}_{\varphi }& X_{14,14}^{\left(2\right)}+X_{10,10}^{\left(3\right)}\\ -{(X_{8,15}^{\left(2\right)}+X_{6,11}^{\left(3\right)})}_{\varphi }& -{(X_{9,15}^{\left(2\right)}+X_{7,11}^{\left(3\right)})}_{\varphi }& -{(X_{10,15}^{\left(2\right)}+X_{8,11}^{\left(3\right)})}_{\varphi }& -{\left(X_{11,15}^{\left(2\right)}\right)}_{\varphi }& -{(X_{13,15}^{\left(2\right)}+X_{9,11}^{\left(3\right)})}_{\varphi }& -{(X_{14,15}^{\left(2\right)}+X_{10,11}^{\left(3\right)})}_{\varphi }\\ -{(X_{8,16}^{\left(2\right)}+X_{6,12}^{\left(3\right)})}_{\varphi }& -{(X_{9,16}^{\left(2\right)}+X_{7,12}^{\left(3\right)})}_{\varphi }& -{(X_{10,16}^{\left(2\right)}+X_{8,12}^{\left(3\right)})}_{\varphi }& -{\left(X_{11,16}^{\left(2\right)}\right)}_{\varphi }& -{(X_{13,16}^{\left(2\right)}+X_{9,12}^{\left(3\right)})}_{\varphi }& -{(X_{14,16}^{\left(2\right)}+X_{10,12}^{\left(3\right)})}_{\varphi }\\ -{\left(X_{8,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{9,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{10,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{11,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{13,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{14,17}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{6,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{7,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{8,13}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{9,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{10,13}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{6,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{7,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{8,14}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{9,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{10,14}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{6,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{7,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{8,15}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{9,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{10,15}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{6,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{7,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{8,16}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{9,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{10,16}^{\left(3\right)}\right)}_{\varphi }\\ 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill D_3^{\left(2\right)}=\left(\begin{array}{cccccccc}{X}_{1,15}^{\left(2\right)}+X_{1,11}^{\left(3\right)}& X_{1,16}^{\left(2\right)}+X_{1,12}^{\left(3\right)}& X_{1,17}^{\left(2\right)}& X_{1,13}^{\left(3\right)}& X_{1,14}^{\left(3\right)}& X_{1,15}^{\left(3\right)}& X_{1,16}^{\left(3\right)}& 0\\ X_{2,15}^{\left(2\right)}+X_{2,11}^{\left(3\right)}& X_{2,16}^{\left(2\right)}+X_{2,12}^{\left(3\right)}& X_{2,17}^{\left(2\right)}& X_{2,13}^{\left(3\right)}& X_{2,14}^{\left(3\right)}& X_{2,15}^{\left(3\right)}& X_{2,16}^{\left(3\right)}& 0\\ X_{3,15}^{\left(2\right)}+X_{3,11}^{\left(3\right)}& X_{3,16}^{\left(2\right)}+X_{3,12}^{\left(3\right)}& X_{3,17}^{\left(2\right)}& X_{3,13}^{\left(3\right)}& X_{3,14}^{\left(3\right)}& X_{3,15}^{\left(3\right)}& X_{3,16}^{\left(3\right)}& 0\\ X_{4,15}^{\left(2\right)}+X_{4,11}^{\left(3\right)}& X_{4,16}^{\left(2\right)}+X_{4,12}^{\left(3\right)}& X_{4,17}^{\left(2\right)}& X_{4,13}^{\left(3\right)}& X_{4,14}^{\left(3\right)}& X_{4,15}^{\left(3\right)}& X_{4,16}^{\left(3\right)}& 0\\ X_{5,15}^{\left(2\right)}& X_{5,16}^{\left(2\right)}& X_{5,17}^{\left(2\right)}& 0& 0& 0& 0& 0\\ X_{7,15}^{\left(2\right)}+X_{5,11}^{\left(3\right)}& X_{7,16}^{\left(2\right)}+X_{5,12}^{\left(3\right)}& X_{7,17}^{\left(2\right)}& X_{5,13}^{\left(3\right)}& X_{5,14}^{\left(3\right)}& X_{5,15}^{\left(3\right)}& X_{5,16}^{\left(3\right)}& 0\\ X_{8,15}^{\left(2\right)}+X_{6,11}^{\left(3\right)}& X_{8,16}^{\left(2\right)}+X_{6,12}^{\left(3\right)}& X_{8,17}^{\left(2\right)}& X_{6,13}^{\left(3\right)}& X_{6,14}^{\left(3\right)}& X_{6,15}^{\left(3\right)}& X_{6,16}^{\left(3\right)}& 0\\ X_{9,15}^{\left(2\right)}+X_{7,11}^{\left(3\right)}& X_{9,16}^{\left(2\right)}+X_{7,12}^{\left(3\right)}& X_{9,17}^{\left(2\right)}& X_{7,13}^{\left(3\right)}& X_{7,14}^{\left(3\right)}& X_{7,15}^{\left(3\right)}& X_{7,16}^{\left(3\right)}& 0\\ X_{10,15}^{\left(2\right)}+X_{8,11}^{\left(3\right)}& X_{10,16}^{\left(2\right)}+X_{8,12}^{\left(3\right)}& X_{10,17}^{\left(2\right)}& X_{8,13}^{\left(3\right)}& X_{8,14}^{\left(3\right)}& X_{8,15}^{\left(3\right)}& X_{8,16}^{\left(3\right)}& 0\\ X_{11,15}^{\left(2\right)}& X_{11,16}^{\left(2\right)}& X_{11,17}^{\left(2\right)}& 0& 0& 0& 0& 0\\ X_{13,15}^{\left(2\right)}+X_{9,11}^{\left(3\right)}& X_{13,16}^{\left(2\right)}+X_{9,12}^{\left(3\right)}& X_{13,17}^{\left(2\right)}& X_{9,13}^{\left(3\right)}& X_{9,14}^{\left(3\right)}& X_{9,15}^{\left(3\right)}& X_{9,16}^{\left(3\right)}& 0\\ X_{14,15}^{\left(2\right)}+X_{10,11}^{\left(3\right)}& X_{14,16}^{\left(2\right)}+X_{10,12}^{\left(3\right)}& X_{14,17}^{\left(2\right)}& X_{10,13}^{\left(3\right)}& X_{10,14}^{\left(3\right)}& X_{10,15}^{\left(3\right)}& X_{10,16}^{\left(3\right)}& 0\\ X_{15,15}^{\left(2\right)}+X_{11,11}^{\left(3\right)}& X_{15,16}^{\left(2\right)}+X_{11,12}^{\left(3\right)}& X_{15,17}^{\left(2\right)}& X_{11,13}^{\left(3\right)}& X_{11,14}^{\left(3\right)}& X_{11,15}^{\left(3\right)}& X_{11,16}^{\left(3\right)}& 0\\ -{(X_{15,16}^{\left(2\right)}+X_{11,12}^{\left(3\right)})}_{\varphi }& X_{16,16}^{\left(2\right)}+X_{12,12}^{\left(3\right)}& X_{16,17}^{\left(2\right)}& X_{12,13}^{\left(3\right)}& X_{12,14}^{\left(3\right)}& X_{12,15}^{\left(3\right)}& X_{12,16}^{\left(3\right)}& 0\\ -{\left(X_{15,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{16,17}^{\left(2\right)}\right)}_{\varphi }& X_{17,17}^{\left(2\right)}& 0& 0& 0& 0& 0\\ -{\left(X_{11,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{12,13}^{\left(3\right)}\right)}_{\varphi }& 0& X_{13,13}^{\left(3\right)}& X_{13,14}^{\left(3\right)}& X_{13,15}^{\left(3\right)}& X_{13,16}^{\left(3\right)}& 0\\ -{\left(X_{11,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{12,14}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{13,14}^{\left(3\right)}\right)}_{\varphi }& X_{14,14}^{\left(3\right)}& X_{14,15}^{\left(3\right)}& X_{14,16}^{\left(3\right)}& 0\\ -{\left(X_{11,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{12,15}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{13,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{14,15}^{\left(3\right)}\right)}_{\varphi }& X_{15,15}^{\left(3\right)}& X_{15,16}^{\left(3\right)}& 0\\ -{\left(X_{11,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{12,16}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{13,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{14,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{15,16}^{\left(3\right)}\right)}_{\varphi }& X_{16,16}^{\left(3\right)}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill D_1^{\left(3\right)}=\left(\begin{array}{cccccc}{X}_{11}^{\left(3\right)}+X_{11}^{\left(4\right)}& X_{12}^{\left(3\right)}+X_{12}^{\left(4\right)}& X_{13}^{\left(3\right)}& X_{15}^{\left(3\right)}+X_{13}^{\left(4\right)}& X_{16}^{\left(3\right)}+X_{14}^{\left(4\right)}& X_{17}^{\left(3\right)}\\ -{(X_{12}^{\left(3\right)}+X_{12}^{\left(4\right)})}_{\varphi }& X_{22}^{\left(3\right)}+X_{22}^{\left(4\right)}& X_{23}^{\left(3\right)}& X_{25}^{\left(3\right)}+X_{23}^{\left(4\right)}& X_{26}^{\left(3\right)}+X_{24}^{\left(4\right)}& X_{27}^{\left(3\right)}\\ -{\left(X_{13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{23}^{\left(3\right)}\right)}_{\varphi }& X_{33}^{\left(3\right)}& X_{35}^{\left(3\right)}& X_{36}^{\left(3\right)}& X_{37}^{\left(3\right)}\\ -{(X_{15}^{\left(3\right)}+X_{13}^{\left(4\right)})}_{\varphi }& -{(X_{25}^{\left(3\right)}+X_{23}^{\left(4\right)})}_{\varphi }& -{\left(X_{35}^{\left(3\right)}\right)}_{\varphi }& X_{55}^{\left(3\right)}+X_{33}^{\left(4\right)}& X_{56}^{\left(3\right)}+X_{34}^{\left(4\right)}& X_{57}^{\left(3\right)}\\ -{(X_{16}^{\left(3\right)}+X_{14}^{\left(4\right)})}_{\varphi }& -{(X_{26}^{\left(3\right)}+X_{24}^{\left(4\right)})}_{\varphi }& -{\left(X_{36}^{\left(3\right)}\right)}_{\varphi }& -{(X_{56}^{\left(3\right)}+X_{34}^{\left(4\right)})}_{\varphi }& X_{66}^{\left(3\right)}+X_{44}^{\left(4\right)}& X_{67}^{\left(3\right)}\\ -{\left(X_{17}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{27}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{37}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{57}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{67}^{\left(3\right)}\right)}_{\varphi }& X_{77}^{\left(3\right)}\\ -{(X_{19}^{\left(3\right)}+X_{15}^{\left(4\right)})}_{\varphi }& -{(X_{29}^{\left(3\right)}+X_{25}^{\left(4\right)})}_{\varphi }& -{\left(X_{39}^{\left(3\right)}\right)}_{\varphi }& -{(X_{59}^{\left(3\right)}+X_{35}^{\left(4\right)})}_{\varphi }& -{(X_{69}^{\left(3\right)}+X_{45}^{\left(4\right)})}_{\varphi }& -{\left(X_{79}^{\left(3\right)}\right)}_{\varphi }\\ -{(X_{1,10}^{\left(3\right)}+X_{16}^{\left(4\right)})}_{\varphi }& -{(X_{2,10}^{\left(3\right)}+X_{26}^{\left(4\right)})}_{\varphi }& -{\left(X_{3,10}^{\left(3\right)}\right)}_{\varphi }& -{(X_{5,10}^{\left(3\right)}+X_{36}^{\left(4\right)})}_{\varphi }& -{(X_{6,10}^{\left(3\right)}+X_{46}^{\left(4\right)})}_{\varphi }& -{\left(X_{7,10}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{1,11}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,11}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{3,11}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{5,11}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{6,11}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{7,11}^{\left(3\right)}\right)}_{\varphi }\\ -{(X_{1,13}^{\left(3\right)}+X_{17}^{\left(4\right)})}_{\varphi }& -{(X_{2,13}^{\left(3\right)}+X_{27}^{\left(4\right)})}_{\varphi }& -{\left(X_{3,13}^{\left(3\right)}\right)}_{\varphi }& -{(X_{5,13}^{\left(3\right)}+X_{37}^{\left(4\right)})}_{\varphi }& -{(X_{6,13}^{\left(3\right)}+X_{47}^{\left(4\right)})}_{\varphi }& -{\left(X_{7,13}^{\left(3\right)}\right)}_{\varphi }\\ -{(X_{1,14}^{\left(3\right)}+X_{18}^{\left(4\right)})}_{\varphi }& -{(X_{2,14}^{\left(3\right)}+X_{28}^{\left(4\right)})}_{\varphi }& -{\left(X_{3,14}^{\left(3\right)}\right)}_{\varphi }& -{(X_{5,14}^{\left(3\right)}+X_{38}^{\left(4\right)})}_{\varphi }& -{(X_{6,14}^{\left(3\right)}+X_{48}^{\left(4\right)})}_{\varphi }& -{\left(X_{7,14}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{1,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{3,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{5,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{6,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{7,15}^{\left(3\right)}\right)}_{\varphi }\\ -{(X_{1,17}^{\left(3\right)}+X_{19}^{\left(4\right)})}_{\varphi }& -{(X_{2,17}^{\left(3\right)}+X_{29}^{\left(4\right)})}_{\varphi }& -{\left(X_{3,17}^{\left(3\right)}\right)}_{\varphi }& -{(X_{5,17}^{\left(3\right)}+X_{39}^{\left(4\right)})}_{\varphi }& -{(X_{6,17}^{\left(3\right)}+X_{49}^{\left(4\right)})}_{\varphi }& -{\left(X_{7,17}^{\left(3\right)}\right)}_{\varphi }\\ -{(X_{1,18}^{\left(3\right)}+X_{1,10}^{\left(4\right)})}_{\varphi }& -{(X_{2,18}^{\left(3\right)}+X_{2,10}^{\left(4\right)})}_{\varphi }& -{\left(X_{3,18}^{\left(3\right)}\right)}_{\varphi }& -{(X_{5,18}^{\left(3\right)}+X_{3,10}^{\left(4\right)})}_{\varphi }& -{(X_{6,18}^{\left(3\right)}+X_{4,10}^{\left(4\right)})}_{\varphi }& -{\left(X_{7,18}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{1,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{3,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{5,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{6,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{7,19}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{1,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{2,11}^{\left(4\right)}\right)}_{\varphi }& 0& -{\left(X_{3,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{4,11}^{\left(4\right)}\right)}_{\varphi }& 0\\ -{\left(X_{1,12}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{2,12}^{\left(4\right)}\right)}_{\varphi }& 0& -{\left(X_{3,12}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{4,12}^{\left(4\right)}\right)}_{\varphi }& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill D_2^{\left(3\right)}=\left(\begin{array}{cccccc}{X}_{19}^{\left(3\right)}+X_{15}^{\left(4\right)}& X_{1,10}^{\left(3\right)}+X_{16}^{\left(4\right)}& X_{1,11}^{\left(3\right)}& X_{1,13}^{\left(3\right)}+X_{17}^{\left(4\right)}& X_{1,14}^{\left(3\right)}+X_{18}^{\left(4\right)}& X_{1,15}^{\left(3\right)}\\ X_{29}^{\left(3\right)}+X_{25}^{\left(4\right)}& X_{2,10}^{\left(3\right)}+X_{26}^{\left(4\right)}& X_{2,11}^{\left(3\right)}& X_{2,13}^{\left(3\right)}+X_{27}^{\left(4\right)}& X_{2,14}^{\left(3\right)}+X_{28}^{\left(4\right)}& X_{2,15}^{\left(3\right)}\\ X_{39}^{\left(3\right)}& X_{3,10}^{\left(3\right)}& X_{3,11}^{\left(3\right)}& X_{3,13}^{\left(3\right)}& X_{3,14}^{\left(3\right)}& X_{3,15}^{\left(3\right)}\\ X_{59}^{\left(3\right)}+X_{35}^{\left(4\right)}& X_{5,10}^{\left(3\right)}+X_{36}^{\left(4\right)}& X_{5,11}^{\left(3\right)}& X_{5,13}^{\left(3\right)}+X_{37}^{\left(4\right)}& X_{5,14}^{\left(3\right)}+X_{38}^{\left(4\right)}& X_{5,15}^{\left(3\right)}\\ X_{69}^{\left(3\right)}+X_{45}^{\left(4\right)}& X_{6,10}^{\left(3\right)}+X_{46}^{\left(4\right)}& X_{6,11}^{\left(3\right)}& X_{6,13}^{\left(3\right)}+X_{47}^{\left(4\right)}& X_{6,14}^{\left(3\right)}+X_{48}^{\left(4\right)}& X_{6,15}^{\left(3\right)}\\ X_{79}^{\left(3\right)}& X_{7,10}^{\left(3\right)}& X_{7,11}^{\left(3\right)}& X_{7,13}^{\left(3\right)}& X_{7,14}^{\left(3\right)}& X_{7,15}^{\left(3\right)}\\ X_{99}^{\left(3\right)}+X_{55}^{\left(4\right)}& X_{9,10}^{\left(3\right)}+X_{56}^{\left(4\right)}& X_{9,11}^{\left(3\right)}& X_{9,13}^{\left(3\right)}+X_{57}^{\left(4\right)}& X_{9,14}^{\left(3\right)}+X_{58}^{\left(4\right)}& X_{9,15}^{\left(3\right)}\\ -{(X_{9,10}^{\left(3\right)}+X_{56}^{\left(4\right)})}_{\varphi }& X_{10,10}^{\left(3\right)}+X_{66}^{\left(4\right)}& X_{10,11}^{\left(3\right)}& X_{10,13}^{\left(3\right)}+X_{67}^{\left(4\right)}& X_{10,14}^{\left(3\right)}+X_{68}^{\left(4\right)}& X_{10,15}^{\left(3\right)}\\ -{\left(X_{9,11}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{10,11}^{\left(3\right)}\right)}_{\varphi }& X_{11,11}^{\left(3\right)}& X_{11,13}^{\left(3\right)}& X_{11,14}^{\left(3\right)}& X_{11,15}^{\left(3\right)}\\ -{(X_{9,13}^{\left(3\right)}+X_{57}^{\left(4\right)})}_{\varphi }& -{(X_{10,13}^{\left(3\right)}+X_{67}^{\left(4\right)})}_{\varphi }& -{\left(X_{11,13}^{\left(3\right)}\right)}_{\varphi }& X_{13,13}^{\left(3\right)}+X_{77}^{\left(4\right)}& X_{13,14}^{\left(3\right)}+X_{78}^{\left(4\right)}& X_{13,15}^{\left(3\right)}\\ -{(X_{9,14}^{\left(3\right)}+X_{58}^{\left(4\right)})}_{\varphi }& -{(X_{10,14}^{\left(3\right)}+X_{68}^{\left(4\right)})}_{\varphi }& -{\left(X_{11,14}^{\left(3\right)}\right)}_{\varphi }& -{(X_{13,14}^{\left(3\right)}+X_{78}^{\left(4\right)})}_{\varphi }& X_{14,14}^{\left(3\right)}+X_{88}^{\left(4\right)}& X_{14,15}^{\left(3\right)}\\ -{\left(X_{9,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{10,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{11,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{13,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{14,15}^{\left(3\right)}\right)}_{\varphi }& X_{15,15}^{\left(3\right)}\\ -{(X_{9,17}^{\left(3\right)}+X_{59}^{\left(4\right)})}_{\varphi }& -{(X_{10,17}^{\left(3\right)}+X_{69}^{\left(4\right)})}_{\varphi }& -{\left(X_{11,17}^{\left(3\right)}\right)}_{\varphi }& -{(X_{13,17}^{\left(3\right)}+X_{79}^{\left(4\right)})}_{\varphi }& -{(X_{14,17}^{\left(3\right)}+X_{89}^{\left(4\right)})}_{\varphi }& -{\left(X_{15,17}^{\left(3\right)}\right)}_{\varphi }\\ -{(X_{9,18}^{\left(3\right)}+X_{5,10}^{\left(4\right)})}_{\varphi }& -{(X_{10,18}^{\left(3\right)}+X_{6,10}^{\left(4\right)})}_{\varphi }& -{\left(X_{11,18}^{\left(3\right)}\right)}_{\varphi }& -{(X_{13,18}^{\left(3\right)}+X_{7,10}^{\left(4\right)})}_{\varphi }& -{(X_{14,18}^{\left(3\right)}+X_{8,10}^{\left(4\right)})}_{\varphi }& -{\left(X_{15,18}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{9,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{10,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{11,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{13,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{14,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{15,19}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{5,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{6,11}^{\left(4\right)}\right)}_{\varphi }& 0& -{\left(X_{7,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{8,11}^{\left(4\right)}\right)}_{\varphi }& 0\\ -{\left(X_{5,12}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{6,12}^{\left(4\right)}\right)}_{\varphi }& 0& -{\left(X_{7,12}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{8,12}^{\left(4\right)}\right)}_{\varphi }& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{c}\hfill D_3^{\left(3\right)}=\left(\begin{array}{cccccc}{X}_{1,17}^{\left(3\right)}+X_{19}^{\left(4\right)}& X_{1,18}^{\left(3\right)}+X_{1,10}^{\left(4\right)}& X_{1,19}^{\left(3\right)}& X_{1,11}^{\left(4\right)}& X_{1,12}^{\left(4\right)}& 0\\ X_{2,17}^{\left(3\right)}+X_{29}^{\left(4\right)}& X_{2,18}^{\left(3\right)}+X_{2,10}^{\left(4\right)}& X_{2,19}^{\left(3\right)}& X_{2,11}^{\left(4\right)}& X_{2,12}^{\left(4\right)}& 0\\ X_{3,17}^{\left(3\right)}& X_{3,18}^{\left(3\right)}& X_{3,19}^{\left(3\right)}& 0& 0& 0\\ X_{5,17}^{\left(3\right)}+X_{39}^{\left(4\right)}& X_{5,18}^{\left(3\right)}+X_{3,10}^{\left(4\right)}& X_{5,19}^{\left(3\right)}& X_{3,11}^{\left(4\right)}& X_{3,12}^{\left(4\right)}& 0\\ X_{6,17}^{\left(3\right)}+X_{49}^{\left(4\right)}& X_{6,18}^{\left(3\right)}+X_{4,10}^{\left(4\right)}& X_{6,19}^{\left(3\right)}& X_{4,11}^{\left(4\right)}& X_{4,12}^{\left(4\right)}& 0\\ X_{7,17}^{\left(3\right)}& X_{7,18}^{\left(3\right)}& X_{7,19}^{\left(3\right)}& 0& 0& 0\\ X_{9,17}^{\left(3\right)}+X_{59}^{\left(4\right)}& X_{9,18}^{\left(3\right)}+X_{5,10}^{\left(4\right)}& X_{9,19}^{\left(3\right)}& X_{5,11}^{\left(4\right)}& X_{5,12}^{\left(4\right)}& 0\\ X_{10,17}^{\left(3\right)}+X_{69}^{\left(4\right)}& X_{10,18}^{\left(3\right)}+X_{6,10}^{\left(4\right)}& X_{10,19}^{\left(3\right)}& X_{6,11}^{\left(4\right)}& X_{6,12}^{\left(4\right)}& 0\\ X_{11,17}^{\left(3\right)}& X_{11,18}^{\left(3\right)}& X_{11,19}^{\left(3\right)}& 0& 0& 0\\ X_{13,17}^{\left(3\right)}+X_{79}^{\left(4\right)}& X_{13,18}^{\left(3\right)}+X_{7,10}^{\left(4\right)}& X_{13,19}^{\left(3\right)}& X_{7,11}^{\left(4\right)}& X_{7,12}^{\left(4\right)}& 0\\ X_{14,17}^{\left(3\right)}+X_{89}^{\left(4\right)}& X_{14,18}^{\left(3\right)}+X_{8,10}^{\left(4\right)}& X_{14,19}^{\left(3\right)}& X_{8,11}^{\left(4\right)}& X_{8,12}^{\left(4\right)}& 0\\ X_{15,17}^{\left(3\right)}& X_{15,18}^{\left(3\right)}& X_{15,19}^{\left(3\right)}& 0& 0& 0\\ X_{17,17}^{\left(3\right)}+X_{99}^{\left(4\right)}& X_{17,18}^{\left(3\right)}+X_{9,10}^{\left(4\right)}& X_{17,19}^{\left(3\right)}& X_{9,11}^{\left(4\right)}& X_{9,12}^{\left(4\right)}& 0\\ -{(X_{17,18}^{\left(3\right)}+X_{9,10}^{\left(4\right)})}_{\varphi }& X_{18,18}^{\left(3\right)}+X_{10,10}^{\left(4\right)}& X_{18,19}^{\left(3\right)}& X_{10,11}^{\left(4\right)}& X_{10,12}^{\left(4\right)}& 0\\ -{\left(X_{17,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{18,19}^{\left(3\right)}\right)}_{\varphi }& X_{19,19}^{\left(3\right)}& 0& 0& 0\\ -{\left(X_{9,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{10,11}^{\left(4\right)}\right)}_{\varphi }& 0& X_{11,11}^{\left(4\right)}& X_{11,12}^{\left(4\right)}& 0\\ -{\left(X_{9,12}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{10,12}^{\left(4\right)}\right)}_{\varphi }& 0& -{\left(X_{11,12}^{\left(4\right)}\right)}_{\varphi }& X_{12,12}^{\left(4\right)}& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right).\end{array}$$

Observe that the matrices (11)–(14) are all defined in [20].

It follows from (11)–(14) that the general $\varphi$-skew-Hermitian solution to the system (4) is provided in the form (7)–(10). □

Example 1.

Given a system (1). We consider the general ϕ-skew-Hermitian solution to this system, where $\varphi \left(\mathit{a}\right)={\mathit{a}}^{\mathrm{j}\ast }=-\mathbf{j}{\mathit{a}}^{\ast }\mathbf{j}$ for $\mathit{a}\in \mathbb{H}$. The quaternion matrices $A_i$, $B_i$, (i = 1, 2, 3), $A_4$ are given:

$$A_1=\left(\begin{array}{ccc}\mathbf{i}& 0& 1+\mathbf{k}\\ 0& \mathbf{j}-\mathbf{k}& \mathbf{j}+\mathbf{k}\\ 1& 0& \mathbf{k}\end{array}\right),A_2=\left(\begin{array}{ccc}\mathbf{j}& 2\mathbf{i}& 0\\ 3\mathbf{k}& 0& \mathbf{i}+\mathbf{j}\\ 5+\mathbf{j}& 6& 0\end{array}\right),$$

$$A_3=\left(\begin{array}{ccc}\mathbf{i}& 0& 0\\ 0& \mathbf{i}+\mathbf{j}& 2\mathbf{i}\\ 2\mathbf{i}+\mathbf{k}& \mathbf{k}& 0\end{array}\right),A_4=\left(\begin{array}{ccc}0& \mathbf{i}+\mathbf{j}& 0\\ \mathbf{k}& 0& \mathbf{i}+\mathbf{j}-\mathbf{k}\\ 2\mathbf{j}& 0& \mathbf{i}+3\mathbf{k}\end{array}\right),$$

$$B_1=\left(\begin{array}{ccc}1+\mathbf{i}& 3& 0\\ 0& \mathbf{i}+\mathbf{k}& \mathbf{j}-\mathbf{k}\\ \mathbf{j}& 0& 2+\mathbf{k}\end{array}\right),B_2=\left(\begin{array}{ccc}0& \mathbf{k}& 2\mathbf{j}\\ 3\mathbf{i}& \mathbf{i}-\mathbf{j}& 0\\ 5\mathbf{k}& 0& \mathbf{j}+\mathbf{k}\end{array}\right),B_3=\left(\begin{array}{ccc}0& \mathbf{k}& \mathbf{i}+\mathbf{j}\\ 0& 2\mathbf{i}& 0\\ 3\mathbf{j}& 0& 5\mathbf{k}\end{array}\right).$$

The ϕ-skew-Hermitian matrices $C_i$, (i = 1,2,3,4) are given:

$$C_1=\left(\begin{array}{ccc}4\mathbf{j}& 6+11\mathbf{i}+3\mathbf{j}& -9+8\mathbf{i}+16\mathbf{j}+23\mathbf{k}\\ -6-11\mathbf{i}+3\mathbf{j}& 18\mathbf{j}& -24-10\mathbf{i}-12\mathbf{j}+2\mathbf{k}\\ 9-8\mathbf{i}+16\mathbf{j}-23\mathbf{k}& 24+10\mathbf{i}-12\mathbf{j}-2\mathbf{k}& 15\mathbf{j}\end{array}\right),$$

$$C_2=\left(\begin{array}{ccc}-15\mathbf{j}& 6+2\mathbf{i}+20\mathbf{j}+3\mathbf{k}& -10-10\mathbf{i}+3\mathbf{j}-14\mathbf{k}\\ -6-2\mathbf{i}+20\mathbf{j}-3\mathbf{k}& 45\mathbf{j}& -36-50\mathbf{i}+62\mathbf{j}+39\mathbf{k}\\ 10+10\mathbf{i}+3\mathbf{j}+14\mathbf{k}& 36+50\mathbf{i}+62\mathbf{j}-39\mathbf{k}& 43\mathbf{j}\end{array}\right),$$

$$C_3=\left(\begin{array}{ccc}4\mathbf{j}& -2-3\mathbf{i}+\mathbf{j}& -12-4\mathbf{j}-4\mathbf{k}\\ 2+3\mathbf{i}+\mathbf{j}& -14\mathbf{j}& 6+18\mathbf{i}-4\mathbf{j}\\ 12-4\mathbf{j}+4\mathbf{k}& -6-18\mathbf{i}-4\mathbf{j}& 39\mathbf{j}\end{array}\right),$$

$$C_4=\left(\begin{array}{ccc}-2\mathbf{j}& -2+2\mathbf{i}+2\mathbf{j}& 4+4\mathbf{i}+4\mathbf{k}\\ 2-2\mathbf{i}+2\mathbf{j}& 9\mathbf{j}& -6-3\mathbf{i}-2\mathbf{j}-5\mathbf{k}\\ -4-4\mathbf{i}-4\mathbf{k}& 6+3\mathbf{i}-2\mathbf{j}+5\mathbf{k}& 12\mathbf{j}\end{array}\right).$$

According to Theorem 1, the following ϕ-skew-Hermitian matrices satisfy the system

$$X_1=-{\left(X_1\right)}_{\varphi }=\left(\begin{array}{ccc}\mathbf{j}& 2+7\mathbf{k}& 3\mathbf{i}+\mathbf{j}-\mathbf{k}\\ -2-7\mathbf{k}& 5\mathbf{j}& \mathbf{i}+\mathbf{k}\\ -3\mathbf{i}+\mathbf{j}+\mathbf{k}& -\mathbf{i}-\mathbf{k}& 3\mathbf{j}\end{array}\right),$$

$$X_2=-{\left(X_2\right)}_{\varphi }=\left(\begin{array}{ccc}2\mathbf{j}& 1+2\mathbf{i}& 0\\ -1-2\mathbf{i}& -\mathbf{j}& 2\mathbf{j}+3\mathbf{k}\\ 0& 2\mathbf{j}-3\mathbf{k}& 3\mathbf{j}\end{array}\right),$$

$$X_3=-{\left(X_3\right)}_{\varphi }=\left(\begin{array}{ccc}\mathbf{j}& \mathbf{j}& 2\mathbf{i}\\ \mathbf{j}& 3\mathbf{j}& \mathbf{j}+3\mathbf{k}\\ -2\mathbf{i}& \mathbf{j}-3\mathbf{k}& -5\mathbf{j}\end{array}\right),X_4=-{\left(X_4\right)}_{\varphi }=\left(\begin{array}{ccc}\mathbf{j}& \mathbf{i}+\mathbf{j}+2\mathbf{k}& \mathbf{k}\\ -\mathbf{i}+\mathbf{j}-2\mathbf{k}& -\mathbf{j}& 0\\ -\mathbf{k}& 0& 2\mathbf{j}\end{array}\right).$$

4. The β(ϕ)-Signature Bounds of the Solution X₁ to the System (1)

In this section, we investigate the property of the solution $X_1$ to the system (1). Firstly, we consider the $\beta$($\varphi$)-signature bounds of the $\varphi$-skew-Hermitian solution $X_1$ to the system (1). The following Lemmas provide the $\beta$($\varphi$)-signature bounds and minimum rank of block matrices.

Lemma 2

([19]). Let M be a ϕ-skew-Hermitian block matrix

$$\begin{array}{c}\hfill M=\left(\begin{array}{cc}X& A\\ -A_{\varphi }& B\end{array}\right),\end{array}$$

(15)

where $A\in {\mathbb{H}}^{n\times m}$ and $B=-B_{\varphi }\in {\mathbb{H}}^{m\times m}$ are given, $X\in {\mathbb{H}}^{n\times n}$ is a variable ϕ-skew-Hermitian matrix. Then,

$$\underset{X=-X_{\varphi }}{max}{ln}_{\pm }\left(M\right)=n+{ln}_{\pm }\left(B\right),\underset{X=-X_{\varphi }}{min}{ln}_{\pm }\left(M\right)=r\left(\begin{array}{c}A\\ B\end{array}\right)-{ln}_{\mp }\left(B\right).$$

Lemma 3

([26,27,28,29]). Let N be a block matrix

$$\begin{array}{c}\hfill N=\left(\begin{array}{cc}A& B\\ D& Y\end{array}\right),\end{array}$$

(16)

where A, B and D are given quaternion matrices, $Y\in {\mathbb{H}}^{n\times m}$ is a variable matrix. Then,

$$\underset{Y}{min}r\left(N\right)=r\left(\begin{array}{cc}A& B\end{array}\right)+r\left(\begin{array}{c}A\\ D\end{array}\right)-r\left(A\right).$$

The following Theorem derives the $\beta$($\varphi$)-signature bounds of the solution $X_1$ to the system (1).

Theorem 2.

Assume that the system (1) has a ϕ-skew-Hermitian solution $(X_1,X_2,X_3,X_4)\in {\mathbb{H}}^{t_1\times t_1}\times {\mathbb{H}}^{t_2\times t_2}\times {\mathbb{H}}^{t_3\times t_3}\times {\mathbb{H}}^{t_4\times t_4}$. We denote

$$S=\{X_1=-{\left(X_1\right)}_{\varphi }\in {\mathbb{H}}^{t_1\times t_1}\mid A_i{X}_i{\left(A_i\right)}_{\varphi }+B_i{X}_{i+1}{\left(B_i\right)}_{\varphi }=C_i,(i=1,2,3),A_4{X}_4{\left(A_4\right)}_{\varphi }=C_4\}.$$

Then, we can obtain

$$\begin{array}{c}\hfill max{ln}_{\pm }\left(X_1\right)=t_1-r\left(\begin{array}{c}{A}_1\end{array}\right)-r\left(\begin{array}{ccc}{B}_1& 0& 0\\ A_2& B_2& 0\\ 0& A_3& B_3\end{array}\right)-r\left(\begin{array}{ccc}{A}_4& 0& 0\\ B_3& A_3& 0\\ 0& B_2& A_2\\ 0& 0& B_1\end{array}\right)\end{array}$$

$$\begin{array}{c}\hfill +r\left(\begin{array}{cccc}{A}_1& B_1& 0& 0\\ 0& A_2& B_2& 0\\ 0& 0& A_3& B_3\end{array}\right)+{ln}_{\pm }\left(\begin{array}{ccccccc}-C_4& A_4& 0& 0& 0& 0& 0\\ {\left(A_4\right)}_{\varphi }& 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ 0& B_3& C_3& A_3& 0& 0& 0\\ 0& 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& 0& B_1& C_1\end{array}\right),\end{array}$$

(17)

$$\begin{array}{c}\hfill min{ln}_{\pm }\left(X_1\right)=r\left(\begin{array}{c}{\left(B_1\right)}_{\varphi }\\ C_1\end{array}\right)-r\left(\begin{array}{cc}{A}_2& 0\\ 0& {\left(B_1\right)}_{\varphi }\\ B_1& C_1\end{array}\right)+r\left(\begin{array}{ccc}{\left(B_2\right)}_{\varphi }& 0& 0\\ -C_2& A_2& 0\\ {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& B_1& C_1\end{array}\right)\end{array}$$

$$\begin{array}{c}\hfill -r\left(\begin{array}{cccc}{A}_3& 0& 0& 0\\ 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ B_2& -C_2& A_2& 0\\ 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& B_1& C_1\end{array}\right)+r\left(\begin{array}{ccccc}{\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ C_3& A_3& 0& 0& 0\\ {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& B_2& -C_2& A_2& 0\\ 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& B_1& C_1\end{array}\right)\end{array}$$

$$\begin{array}{c}\hfill -r\left(\begin{array}{cccccc}{A}_4& 0& 0& 0& 0& 0\\ 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ B_3& C_3& A_3& 0& 0& 0\\ 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& B_1& C_1\end{array}\right)+{ln}_{\pm }\left(\begin{array}{ccccccc}-C_4& A_4& 0& 0& 0& 0& 0\\ {\left(A_4\right)}_{\varphi }& 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ 0& B_3& C_3& A_3& 0& 0& 0\\ 0& 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& 0& B_1& C_1\end{array}\right).\end{array}$$

(18)

Proof. 

According to Theorem 1, the $\varphi$-skew-Hermitian solution $X_1$ can be written as

$$X_1={\widehat{T}}_1{\widehat{X}}_1{\left({\widehat{T}}_1\right)}_{\varphi },$$

we have

$$\underset{X_1\in S}{{ln}_{\pm }\left(X_1\right)}={ln}_{\pm }\left({\widehat{T}}_1{\widehat{X}}_1{\left({\widehat{T}}_1\right)}_{\varphi }\right)={ln}_{\pm }\left({\widehat{X}}_1\right).$$

Thus, in order to study the $\beta$($\varphi$)-signature bounds of $X_1$ under S, we just have to consider the $\beta$($\varphi$)-signature bounds of ${\widehat{X}}_1$. Assume ${\widehat{X}}_1=({\widehat{X}}_1^{\left(1\right)},{\widehat{X}}_1^{\left(2\right)}),$ where

$${\widehat{X}}_1^{\left(1\right)}=\begin{array}{cc}& \begin{array}{cccccc}\hspace{1em}\hspace{1em}& n_1& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& n_2& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& n_3\end{array}\\ \begin{array}{c}{n}_1\\ n_2\\ n_3\\ n_4\\ n_5\\ n_6\\ n_7\\ t_1-r_{a_1}\end{array}& \left(\begin{array}{ccc}{D}_{11}^{\left(1\right)}-D_{11}^{\left(2\right)}+D_{11}^{\left(3\right)}-D_{11}^{\left(4\right)}& D_{12}^{\left(1\right)}-D_{12}^{\left(2\right)}+D_{12}^{\left(3\right)}-X_{12}^{\left(4\right)}& D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)}\\ -{(D_{12}^{\left(1\right)}-D_{12}^{\left(2\right)}+D_{12}^{\left(3\right)}-X_{12}^{\left(4\right)})}_{\varphi }& D_{22}^{\left(1\right)}-D_{22}^{\left(2\right)}+D_{22}^{\left(3\right)}-X_{22}^{\left(4\right)}& D_{23}^{\left(1\right)}-D_{23}^{\left(2\right)}+D_{23}^{\left(3\right)}\\ -{(D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)})}_{\varphi }& -{(D_{23}^{\left(1\right)}-D_{23}^{\left(2\right)}+D_{23}^{\left(3\right)})}_{\varphi }& D_{33}^{\left(1\right)}-D_{33}^{\left(2\right)}+D_{33}^{\left(3\right)}\\ -{(D_{14}^{\left(1\right)}-D_{14}^{\left(2\right)}+X_{14}^{\left(3\right)})}_{\varphi }& -{(D_{24}^{\left(1\right)}-D_{24}^{\left(2\right)}+X_{24}^{\left(3\right)})}_{\varphi }& -{(D_{34}^{\left(1\right)}-D_{34}^{\left(2\right)}+X_{34}^{\left(3\right)})}_{\varphi }\\ -{(D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)})}_{\varphi }& -{(D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)})}_{\varphi }& -{(D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)})}_{\varphi }\\ -{(D_{16}^{\left(1\right)}-X_{16}^{\left(2\right)})}_{\varphi }& -{(D_{26}^{\left(1\right)}-X_{26}^{\left(2\right)})}_{\varphi }& -{(D_{36}^{\left(1\right)}-X_{36}^{\left(2\right)})}_{\varphi }\\ -{\left(D_{17}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{27}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{37}^{\left(1\right)}\right)}_{\varphi }\\ -{\left(X_{18}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{28}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{38}^{\left(1\right)}\right)}_{\varphi }\end{array}\right)\end{array},$$

$${\widehat{X}}_1^{\left(2\right)}=\begin{array}{cc}& \begin{array}{cccccccccc}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}& n_4& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& n_5& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}& n_6& \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}& n_7& \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}& t_1-r_{a_1}\end{array}\\ \begin{array}{c}{n}_1\\ n_2\\ n_3\\ n_4\\ n_5\\ n_6\\ n_7\\ t_1-r_{a_1}\end{array}& \left(\begin{array}{ccccc}{D}_{14}^{\left(1\right)}-D_{14}^{\left(2\right)}+X_{14}^{\left(3\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}& D_{16}^{\left(1\right)}-X_{16}^{\left(2\right)}& D_{17}^{\left(1\right)}& X_{18}^{\left(1\right)}\\ D_{24}^{\left(1\right)}-D_{24}^{\left(2\right)}+X_{24}^{\left(3\right)}& D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)}& D_{26}^{\left(1\right)}-X_{26}^{\left(2\right)}& D_{27}^{\left(1\right)}& X_{28}^{\left(1\right)}\\ D_{34}^{\left(1\right)}-D_{34}^{\left(2\right)}+X_{34}^{\left(3\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}& D_{36}^{\left(1\right)}-X_{36}^{\left(2\right)}& D_{37}^{\left(1\right)}& X_{38}^{\left(1\right)}\\ D_{44}^{\left(1\right)}-D_{44}^{\left(2\right)}+X_{44}^{\left(3\right)}& D_{45}^{\left(1\right)}-D_{45}^{\left(2\right)}& D_{46}^{\left(1\right)}-X_{46}^{\left(2\right)}& D_{47}^{\left(1\right)}& X_{48}^{\left(1\right)}\\ -{(D_{45}^{\left(1\right)}-D_{45}^{\left(2\right)})}_{\varphi }& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}& D_{56}^{\left(1\right)}-X_{56}^{\left(2\right)}& D_{57}^{\left(1\right)}& X_{58}^{\left(1\right)}\\ -{(D_{46}^{\left(1\right)}-X_{46}^{\left(2\right)})}_{\varphi }& -{(D_{56}^{\left(1\right)}-X_{56}^{\left(2\right)})}_{\varphi }& D_{66}^{\left(1\right)}-X_{66}^{\left(2\right)}& D_{67}^{\left(1\right)}& X_{68}^{\left(1\right)}\\ -{\left(D_{47}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{67}^{\left(1\right)}\right)}_{\varphi }& D_{77}^{\left(1\right)}& X_{78}^{\left(1\right)}\\ -{\left(X_{48}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{58}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{68}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{78}^{\left(1\right)}\right)}_{\varphi }& X_{88}^{\left(1\right)}\end{array}\right)\end{array}.$$

We treat matrix ${\widehat{X}}_1$ as a block matrix, using the Lemma 2 and Lemma 3 to get the $\beta$($\varphi$)-signature bounds of the $\varphi$-skew-Hermitian matrix ${\widehat{X}}_1$, which is equivalent to the $\beta$($\varphi$)-signature bounds of the $\varphi$-skew-Hermitian solution $X_1$. The specific steps are as follows.

Step 1. We treat the variable $\varphi$-skew-Hermitian matrix $X_{88}^{\left(1\right)}$ of ${\widehat{X}}_1$ as the matrix block X in (15). According to Lemma 2, we derive

$$\underset{X_{88}^{\left(1\right)}}{max}{ln}_{\pm }\left({\widehat{X}}_1\right)=t_1-r_{a_1}+{ln}_{\pm }\left({\mathsf{\Psi }}_1\right),\underset{X_{88}^{\left(1\right)}}{min}{ln}_{\pm }\left({\widehat{X}}_1\right)=r\left(\mathsf{\Psi }\right)-{ln}_{\mp }\left({\mathsf{\Psi }}_1\right),$$

assume $\mathsf{\Psi }=({\mathsf{\Psi }}^{\left(1\right)},{\mathsf{\Psi }}^{\left(2\right)})$, ${\mathsf{\Psi }}_1=({\mathsf{\Psi }}_1^{\left(1\right)},{\mathsf{\Psi }}_1^{\left(2\right)})$, where

$${\mathsf{\Psi }}^{\left(1\right)}=\begin{array}{cc}& \begin{array}{cccccc}{n}_7& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& n_1& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& n_2& \hspace{1em}\hspace{1em}\end{array}\\ \begin{array}{c}{n}_7\\ n_1\\ n_2\\ n_3\\ n_4\\ n_5\\ n_6\\ t_1-r_{a_1}\end{array}& \left(\begin{array}{ccc}{D}_{77}^{\left(1\right)}& -{\left(D_{17}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{27}^{\left(1\right)}\right)}_{\varphi }\\ D_{17}^{\left(1\right)}& D_{11}^{\left(1\right)}-D_{11}^{\left(2\right)}+D_{11}^{\left(3\right)}-D_{11}^{\left(4\right)}& D_{12}^{\left(1\right)}-D_{12}^{\left(2\right)}+D_{12}^{\left(3\right)}-X_{12}^{\left(4\right)}\\ D_{27}^{\left(1\right)}& -{(D_{12}^{\left(1\right)}-D_{12}^{\left(2\right)}+D_{12}^{\left(3\right)}-X_{12}^{\left(4\right)})}_{\varphi }& D_{22}^{\left(1\right)}-D_{22}^{\left(2\right)}+D_{22}^{\left(3\right)}-X_{22}^{\left(4\right)}\\ D_{37}^{\left(1\right)}& -{(D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)})}_{\varphi }& -{(D_{23}^{\left(1\right)}-D_{23}^{\left(2\right)}+D_{23}^{\left(3\right)})}_{\varphi }\\ D_{47}^{\left(1\right)}& -{(D_{14}^{\left(1\right)}-D_{14}^{\left(2\right)}+X_{14}^{\left(3\right)})}_{\varphi }& -{(D_{24}^{\left(1\right)}-D_{24}^{\left(2\right)}+X_{24}^{\left(3\right)})}_{\varphi }\\ D_{57}^{\left(1\right)}& -{(D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)})}_{\varphi }& -{(D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)})}_{\varphi }\\ D_{67}^{\left(1\right)}& -{(D_{16}^{\left(1\right)}-X_{16}^{\left(2\right)})}_{\varphi }& -{(D_{26}^{\left(1\right)}-X_{26}^{\left(2\right)})}_{\varphi }\\ {\left(X_{78}^{\left(1\right)}\right)}_{\varphi }& {\left(X_{18}^{\left(1\right)}\right)}_{\varphi }& {\left(X_{28}^{\left(1\right)}\right)}_{\varphi }\end{array}\right)\end{array},$$

$${\mathsf{\Psi }}^{\left(2\right)}=\begin{array}{cc}& \begin{array}{cccccccc}\hspace{1em}\hspace{1em}\hspace{1em}& n_3& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& n_4& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& n_5& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}& n_6\end{array}\\ \begin{array}{c}{n}_7\\ n_1\\ n_2\\ n_3\\ n_4\\ n_5\\ n_6\\ t_1-r_{a_1}\end{array}& \left(\begin{array}{cccc}-{\left(D_{37}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{47}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{67}^{\left(1\right)}\right)}_{\varphi }\\ D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)}& D_{14}^{\left(1\right)}-D_{14}^{\left(2\right)}+X_{14}^{\left(3\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}& D_{16}^{\left(1\right)}-X_{16}^{\left(2\right)}\\ D_{23}^{\left(1\right)}-D_{23}^{\left(2\right)}+D_{23}^{\left(3\right)}& D_{24}^{\left(1\right)}-D_{24}^{\left(2\right)}+X_{24}^{\left(3\right)}& D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)}& D_{26}^{\left(1\right)}-X_{26}^{\left(2\right)}\\ D_{33}^{\left(1\right)}-D_{33}^{\left(2\right)}+D_{33}^{\left(3\right)}& D_{34}^{\left(1\right)}-D_{34}^{\left(2\right)}+X_{34}^{\left(3\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}& D_{36}^{\left(1\right)}-X_{36}^{\left(2\right)}\\ -{(D_{34}^{\left(1\right)}-D_{34}^{\left(2\right)}+X_{34}^{\left(3\right)})}_{\varphi }& D_{44}^{\left(1\right)}-D_{44}^{\left(2\right)}+X_{44}^{\left(3\right)}& D_{45}^{\left(1\right)}-D_{45}^{\left(2\right)}& D_{46}^{\left(1\right)}-X_{46}^{\left(2\right)}\\ -{(D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)})}_{\varphi }& -{(D_{45}^{\left(1\right)}-D_{45}^{\left(2\right)})}_{\varphi }& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}& D_{56}^{\left(1\right)}-X_{56}^{\left(2\right)}\\ -{(D_{36}^{\left(1\right)}-X_{36}^{\left(2\right)})}_{\varphi }& -{(D_{46}^{\left(1\right)}-X_{46}^{\left(2\right)})}_{\varphi }& -{(D_{56}^{\left(1\right)}-X_{56}^{\left(2\right)})}_{\varphi }& D_{66}^{\left(1\right)}-X_{66}^{\left(2\right)}\\ {\left(X_{38}^{\left(1\right)}\right)}_{\varphi }& {\left(X_{48}^{\left(1\right)}\right)}_{\varphi }& {\left(X_{58}^{\left(1\right)}\right)}_{\varphi }& {\left(X_{68}^{\left(1\right)}\right)}_{\varphi }\end{array}\right)\end{array},$$

$${\mathsf{\Psi }}_1^{\left(1\right)}=\begin{array}{cc}& \begin{array}{cccccc}{n}_7& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& n_1& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}& n_2& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\end{array}\\ \begin{array}{c}{n}_7\\ n_1\\ n_2\\ n_3\\ n_4\\ n_5\\ n_6\end{array}& \left(\begin{array}{ccc}{D}_{77}^{\left(1\right)}& -{\left(D_{17}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{27}^{\left(1\right)}\right)}_{\varphi }\\ D_{17}^{\left(1\right)}& D_{11}^{\left(1\right)}-D_{11}^{\left(2\right)}+D_{11}^{\left(3\right)}-D_{11}^{\left(4\right)}& D_{12}^{\left(1\right)}-D_{12}^{\left(2\right)}+D_{12}^{\left(3\right)}-X_{12}^{\left(4\right)}& \\ D_{27}^{\left(1\right)}& -{(D_{12}^{\left(1\right)}-D_{12}^{\left(2\right)}+D_{12}^{\left(3\right)}-X_{12}^{\left(4\right)})}_{\varphi }& D_{22}^{\left(1\right)}-D_{22}^{\left(2\right)}+D_{22}^{\left(3\right)}-X_{22}^{\left(4\right)}& \\ D_{37}^{\left(1\right)}& -{(D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)})}_{\varphi }& -{(D_{23}^{\left(1\right)}-D_{23}^{\left(2\right)}+D_{23}^{\left(3\right)})}_{\varphi }& \\ D_{47}^{\left(1\right)}& -{(D_{14}^{\left(1\right)}-D_{14}^{\left(2\right)}+X_{14}^{\left(3\right)})}_{\varphi }& -{(D_{24}^{\left(1\right)}-D_{24}^{\left(2\right)}+X_{24}^{\left(3\right)})}_{\varphi }& \\ D_{57}^{\left(1\right)}& -{(D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)})}_{\varphi }& -{(D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)})}_{\varphi }& \\ D_{67}^{\left(1\right)}& -{(D_{16}^{\left(1\right)}-X_{16}^{\left(2\right)})}_{\varphi }& -{(D_{26}^{\left(1\right)}-X_{26}^{\left(2\right)})}_{\varphi }\end{array}\right)\end{array},$$

$${\mathsf{\Psi }}_1^{\left(2\right)}=\begin{array}{cc}& \begin{array}{cccccccc}\hspace{1em}\hspace{1em}\hspace{1em}& n_3& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& n_4& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& n_5& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}& n_6\end{array}\\ \begin{array}{c}{n}_7\\ n_1\\ n_2\\ n_3\\ n_4\\ n_5\\ n_6\end{array}& \left(\begin{array}{cccc}-{\left(D_{37}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{47}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{67}^{\left(1\right)}\right)}_{\varphi }\\ D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)}& D_{14}^{\left(1\right)}-D_{14}^{\left(2\right)}+X_{14}^{\left(3\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}& D_{16}^{\left(1\right)}-X_{16}^{\left(2\right)}\\ D_{23}^{\left(1\right)}-D_{23}^{\left(2\right)}+D_{23}^{\left(3\right)}& D_{24}^{\left(1\right)}-D_{24}^{\left(2\right)}+X_{24}^{\left(3\right)}& D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)}& D_{26}^{\left(1\right)}-X_{26}^{\left(2\right)}\\ D_{33}^{\left(1\right)}-D_{33}^{\left(2\right)}+D_{33}^{\left(3\right)}& D_{34}^{\left(1\right)}-D_{34}^{\left(2\right)}+X_{34}^{\left(3\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}& D_{36}^{\left(1\right)}-X_{36}^{\left(2\right)}\\ -{(D_{34}^{\left(1\right)}-D_{34}^{\left(2\right)}+X_{34}^{\left(3\right)})}_{\varphi }& D_{44}^{\left(1\right)}-D_{44}^{\left(2\right)}+X_{44}^{\left(3\right)}& D_{45}^{\left(1\right)}-D_{45}^{\left(2\right)}& D_{46}^{\left(1\right)}-X_{46}^{\left(2\right)}\\ -{(D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)})}_{\varphi }& -{(D_{45}^{\left(1\right)}-D_{45}^{\left(2\right)})}_{\varphi }& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}& D_{56}^{\left(1\right)}-X_{56}^{\left(2\right)}\\ -{(D_{36}^{\left(1\right)}-X_{36}^{\left(2\right)})}_{\varphi }& -{(D_{46}^{\left(1\right)}-X_{46}^{\left(2\right)})}_{\varphi }& -{(D_{56}^{\left(1\right)}-X_{56}^{\left(2\right)})}_{\varphi }& D_{66}^{\left(1\right)}-X_{66}^{\left(2\right)}\end{array}\right)\end{array}.$$

Then, we treat the matrix $\mathsf{\Psi }$ as a block matrix. According to Lemma 3, we have

$$\underset{(X_{78}^{\left(1\right)},X_{18}^{\left(1\right)},X_{28}^{\left(1\right)},X_{38}^{\left(1\right)},X_{48}^{\left(1\right)},X_{58}^{\left(1\right)},X_{68}^{\left(1\right)})}{min}r\left(\mathsf{\Psi }\right)=r\left({\mathsf{\Psi }}_1\right).$$

Thus, we can obtain

$$\underset{X_{88}^{\left(1\right)},\left(\begin{array}{c}{X}_{78}^{\left(1\right)}\\ X_{18}^{\left(1\right)}\\ X_{28}^{\left(1\right)}\\ X_{38}^{\left(1\right)}\\ X_{48}^{\left(1\right)}\\ X_{58}^{\left(1\right)}\\ X_{68}^{\left(1\right)}\end{array}\right)}{max}{ln}_{\pm }\left({\widehat{X}}_1\right)=t_1-r_{a_1}+{ln}_{\pm }\left({\mathsf{\Psi }}_1\right),\underset{X_{88}^{\left(1\right)},\left(\begin{array}{c}{X}_{78}^{\left(1\right)}\\ X_{18}^{\left(1\right)}\\ X_{28}^{\left(1\right)}\\ X_{38}^{\left(1\right)}\\ X_{48}^{\left(1\right)}\\ X_{58}^{\left(1\right)}\\ X_{68}^{\left(1\right)}\end{array}\right)}{min}{ln}_{\pm }\left({\widehat{X}}_1\right)={ln}_{\pm }\left({\mathsf{\Psi }}_1\right).$$

Step 2. We treating the variable $\varphi$-skew-Hermitian matrix $D_{66}^{\left(1\right)}-X_{66}^{\left(2\right)}$ of ${\mathsf{\Psi }}_1$ as the matrix block X in (15). According to Lemma 2, we provide

$$\underset{D_{66}^{\left(1\right)}-X_{66}^{\left(2\right)}}{max}{ln}_{\pm }\left({\mathsf{\Psi }}_1\right)=n_6+{ln}_{\pm }\left({\mathsf{\Psi }}_3\right),\underset{D_{66}^{\left(1\right)}-X_{66}^{\left(2\right)}}{min}{ln}_{\pm }\left({\mathsf{\Psi }}_1\right)=r\left({\mathsf{\Psi }}_2\right)-{ln}_{\mp }\left({\mathsf{\Psi }}_3\right),$$

assume ${\mathsf{\Psi }}_2=({\mathsf{\Psi }}_2^{\left(1\right)},{\mathsf{\Psi }}_2^{\left(2\right)})$, ${\mathsf{\Psi }}_3=({\mathsf{\Psi }}_3^{\left(1\right)},{\mathsf{\Psi }}_3^{\left(2\right)})$, where

$${\mathsf{\Psi }}_2^{\left(1\right)}=\begin{array}{cc}& \begin{array}{ccccccc}\hspace{0.17em}& n_7& \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& n_1& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}& n_2& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\end{array}\\ \begin{array}{c}{n}_7\\ n_1\\ n_2\\ n_3\\ n_5\\ n_4\\ n_6\end{array}& \left(\begin{array}{ccc}{D}_{77}^{\left(1\right)}& -{\left(D_{17}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{27}^{\left(1\right)}\right)}_{\varphi }\\ D_{17}^{\left(1\right)}& D_{11}^{\left(1\right)}-D_{11}^{\left(2\right)}+D_{11}^{\left(3\right)}-D_{11}^{\left(4\right)}& D_{12}^{\left(1\right)}-D_{12}^{\left(2\right)}+D_{12}^{\left(3\right)}-X_{12}^{\left(4\right)}\\ D_{27}^{\left(1\right)}& -{(D_{12}^{\left(1\right)}-D_{12}^{\left(2\right)}+D_{12}^{\left(3\right)}-X_{12}^{\left(4\right)})}_{\varphi }& D_{22}^{\left(1\right)}-D_{22}^{\left(2\right)}+D_{22}^{\left(3\right)}-X_{22}^{\left(4\right)}\\ D_{37}^{\left(1\right)}& -{(D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)})}_{\varphi }& -{(D_{23}^{\left(1\right)}-D_{23}^{\left(2\right)}+D_{23}^{\left(3\right)})}_{\varphi }\\ D_{57}^{\left(1\right)}& -{(D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)})}_{\varphi }& -{(D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)})}_{\varphi }\\ D_{47}^{\left(1\right)}& -{(D_{14}^{\left(1\right)}-D_{14}^{\left(2\right)}+X_{14}^{\left(3\right)})}_{\varphi }& -{(D_{24}^{\left(1\right)}-D_{24}^{\left(2\right)}+X_{24}^{\left(3\right)})}_{\varphi }\\ D_{67}^{\left(1\right)}& -{(D_{16}^{\left(1\right)}-X_{16}^{\left(2\right)})}_{\varphi }& -{(D_{26}^{\left(1\right)}-X_{26}^{\left(2\right)})}_{\varphi }\end{array}\right)\end{array},$$

$${\mathsf{\Psi }}_2^{\left(2\right)}=\begin{array}{cc}& \begin{array}{ccccccc}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}& n_3& \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& n_5& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}& n_4& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\end{array}\\ \begin{array}{c}{n}_7\\ n_1\\ n_2\\ n_3\\ n_5\\ n_4\\ n_6\end{array}& \left(\begin{array}{ccc}-{\left(D_{37}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{47}^{\left(1\right)}\right)}_{\varphi }\\ D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}& D_{14}^{\left(1\right)}-D_{14}^{\left(2\right)}+X_{14}^{\left(3\right)}\\ D_{23}^{\left(1\right)}-D_{23}^{\left(2\right)}+D_{23}^{\left(3\right)}& D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)}& D_{24}^{\left(1\right)}-D_{24}^{\left(2\right)}+X_{24}^{\left(3\right)}\\ D_{33}^{\left(1\right)}-D_{33}^{\left(2\right)}+D_{33}^{\left(3\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}& D_{34}^{\left(1\right)}-D_{34}^{\left(2\right)}+X_{34}^{\left(3\right)}\\ -{(D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)})}_{\varphi }& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}& -{(D_{45}^{\left(1\right)}-D_{45}^{\left(2\right)})}_{\varphi }\\ -{(D_{34}^{\left(1\right)}-D_{34}^{\left(2\right)}+X_{34}^{\left(3\right)})}_{\varphi }& D_{45}^{\left(1\right)}-D_{45}^{\left(2\right)}& D_{44}^{\left(1\right)}-D_{44}^{\left(2\right)}+X_{44}^{\left(3\right)}\\ -{(D_{36}^{\left(1\right)}-X_{36}^{\left(2\right)})}_{\varphi }& -{(D_{56}^{\left(1\right)}-X_{56}^{\left(2\right)})}_{\varphi }& -{(D_{46}^{\left(1\right)}-X_{46}^{\left(2\right)})}_{\varphi }\end{array}\right),\end{array}$$

$${\mathsf{\Psi }}_3^{\left(1\right)}=\begin{array}{cc}& \begin{array}{ccccccc}\hspace{0.17em}& n_7& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& n_1& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}& n_2& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\end{array}\\ \begin{array}{c}{n}_7\\ n_1\\ n_2\\ n_3\\ n_5\\ n_4\end{array}& \left(\begin{array}{ccc}{D}_{77}^{\left(1\right)}& -{\left(D_{17}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{27}^{\left(1\right)}\right)}_{\varphi }\\ D_{17}^{\left(1\right)}& D_{11}^{\left(1\right)}-D_{11}^{\left(2\right)}+D_{11}^{\left(3\right)}-D_{11}^{\left(4\right)}& D_{12}^{\left(1\right)}-D_{12}^{\left(2\right)}+D_{12}^{\left(3\right)}-X_{12}^{\left(4\right)}\\ D_{27}^{\left(1\right)}& -{(D_{12}^{\left(1\right)}-D_{12}^{\left(2\right)}+D_{12}^{\left(3\right)}-X_{12}^{\left(4\right)})}_{\varphi }& D_{22}^{\left(1\right)}-D_{22}^{\left(2\right)}+D_{22}^{\left(3\right)}-X_{22}^{\left(4\right)}\\ D_{37}^{\left(1\right)}& -{(D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)})}_{\varphi }& -{(D_{23}^{\left(1\right)}-D_{23}^{\left(2\right)}+D_{23}^{\left(3\right)})}_{\varphi }\\ D_{57}^{\left(1\right)}& -{(D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)})}_{\varphi }& -{(D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)})}_{\varphi }\\ D_{47}^{\left(1\right)}& -{(D_{14}^{\left(1\right)}-D_{14}^{\left(2\right)}+X_{14}^{\left(3\right)})}_{\varphi }& -{(D_{24}^{\left(1\right)}-D_{24}^{\left(2\right)}+X_{24}^{\left(3\right)})}_{\varphi }\end{array}\right)\end{array},$$

$${\mathsf{\Psi }}_3^{\left(2\right)}=\begin{array}{cc}& \begin{array}{ccccccc}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& n_3& \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{1em}& n_5& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& n_4& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\end{array}\\ \begin{array}{c}{n}_7\\ n_1\\ n_2\\ n_3\\ n_5\\ n_4\end{array}& \left(\begin{array}{ccc}-{\left(D_{37}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{47}^{\left(1\right)}\right)}_{\varphi }\\ D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}& D_{14}^{\left(1\right)}-D_{14}^{\left(2\right)}+X_{14}^{\left(3\right)}\\ D_{23}^{\left(1\right)}-D_{23}^{\left(2\right)}+D_{23}^{\left(3\right)}& D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)}& D_{24}^{\left(1\right)}-D_{24}^{\left(2\right)}+X_{24}^{\left(3\right)}\\ D_{33}^{\left(1\right)}-D_{33}^{\left(2\right)}+D_{33}^{\left(3\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}& D_{34}^{\left(1\right)}-D_{34}^{\left(2\right)}+X_{34}^{\left(3\right)}\\ -{(D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)})}_{\varphi }& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}& -{(D_{45}^{\left(1\right)}-D_{45}^{\left(2\right)})}_{\varphi }\\ -{(D_{34}^{\left(1\right)}-D_{34}^{\left(2\right)}+X_{34}^{\left(3\right)})}_{\varphi }& D_{45}^{\left(1\right)}-D_{45}^{\left(2\right)}& D_{44}^{\left(1\right)}-D_{44}^{\left(2\right)}+X_{44}^{\left(3\right)}\end{array}\right).\end{array}$$

Then, we treat the matrix ${\mathsf{\Psi }}_2$ as a block matrix. According to Lemma 3, we derive

$$\underset{(D_{16}^{\left(1\right)}-X_{16}^{\left(2\right)},D_{26}^{\left(1\right)}-X_{26}^{\left(2\right)},D_{36}^{\left(1\right)}-X_{36}^{\left(2\right)},D_{56}^{\left(1\right)}-X_{56}^{\left(2\right)},D_{46}^{\left(1\right)}-X_{46}^{\left(2\right)})}{min}r\left({\mathsf{\Psi }}_2\right)=r\left({\mathsf{\Psi }}_3\right)+r\left(\begin{array}{c}{D}_{77}^{\left(1\right)}\\ D_{17}^{\left(1\right)}\\ D_{27}^{\left(1\right)}\\ D_{37}^{\left(1\right)}\\ D_{57}^{\left(1\right)}\\ D_{47}^{\left(1\right)}\\ D_{67}^{\left(1\right)}\end{array}\right)-r\left(\begin{array}{c}{D}_{77}^{\left(1\right)}\\ D_{17}^{\left(1\right)}\\ D_{27}^{\left(1\right)}\\ D_{37}^{\left(1\right)}\\ D_{57}^{\left(1\right)}\\ D_{47}^{\left(1\right)}\end{array}\right).$$

Thus, we can provide

$$\underset{D_{66}^{\left(1\right)}-X_{66}^{\left(2\right)},\left(\begin{array}{c}{D}_{16}^{\left(1\right)}-X_{16}^{\left(2\right)}\\ D_{26}^{\left(1\right)}-X_{26}^{\left(2\right)}\\ D_{36}^{\left(1\right)}-X_{36}^{\left(2\right)}\\ D_{56}^{\left(1\right)}-X_{56}^{\left(2\right)}\\ D_{46}^{\left(1\right)}-X_{46}^{\left(2\right)}\end{array}\right)}{max}{ln}_{\pm }\left({\mathsf{\Psi }}_1\right)=n_6+{ln}_{\pm }\left({\mathsf{\Psi }}_3\right),$$

$$\underset{D_{66}^{\left(1\right)}-X_{66}^{\left(2\right)},\left(\begin{array}{c}{D}_{16}^{\left(1\right)}-X_{16}^{\left(2\right)}\\ D_{26}^{\left(1\right)}-X_{26}^{\left(2\right)}\\ D_{36}^{\left(1\right)}-X_{36}^{\left(2\right)}\\ D_{56}^{\left(1\right)}-X_{56}^{\left(2\right)}\\ D_{46}^{\left(1\right)}-X_{46}^{\left(2\right)}\end{array}\right)}{min}{ln}_{\pm }\left({\mathsf{\Psi }}_1\right)={ln}_{\pm }\left({\mathsf{\Psi }}_3\right)+r\left(\begin{array}{c}{D}_{77}^{\left(1\right)}\\ D_{17}^{\left(1\right)}\\ D_{27}^{\left(1\right)}\\ D_{37}^{\left(1\right)}\\ D_{57}^{\left(1\right)}\\ D_{47}^{\left(1\right)}\\ D_{67}^{\left(1\right)}\end{array}\right)-r\left(\begin{array}{c}{D}_{77}^{\left(1\right)}\\ D_{17}^{\left(1\right)}\\ D_{27}^{\left(1\right)}\\ D_{37}^{\left(1\right)}\\ D_{57}^{\left(1\right)}\\ D_{47}^{\left(1\right)}\end{array}\right).$$

Step 3. Using Lemma 2, Lemma 3 and the similar methods in Step 1 and Step 2 (the specific proof process please see Appendix A), we establish

$$max{ln}_{\pm }\left({\widehat{X}}_1\right)=t_1-r_{a_1}+n_6+n_4+n_2$$

$$\begin{array}{c}\hfill +{ln}_{\pm }\left(\begin{array}{cccc}{D}_{77}^{\left(1\right)}& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{37}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{17}^{\left(1\right)}\right)}_{\varphi }\\ D_{57}^{\left(1\right)}& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}& -{(D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)})}_{\varphi }& -{(D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)})}_{\varphi }\\ D_{37}^{\left(1\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}& D_{33}^{\left(1\right)}-D_{33}^{\left(2\right)}+D_{33}^{\left(3\right)}& -{(D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)})}_{\varphi }\\ D_{17}^{\left(1\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}& D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)}& D_{11}^{\left(1\right)}-D_{11}^{\left(2\right)}+D_{11}^{\left(3\right)}-D_{11}^{\left(4\right)}\end{array}\right),\end{array}$$

(19)

$$min{ln}_{\pm }\left({\widehat{X}}_1\right)=r\left(\begin{array}{c}{D}_{77}^{\left(1\right)}\\ D_{17}^{\left(1\right)}\\ D_{27}^{\left(1\right)}\\ D_{37}^{\left(1\right)}\\ D_{57}^{\left(1\right)}\\ D_{47}^{\left(1\right)}\\ D_{67}^{\left(1\right)}\end{array}\right)-r\left(\begin{array}{c}{D}_{77}^{\left(1\right)}\\ D_{17}^{\left(1\right)}\\ D_{27}^{\left(1\right)}\\ D_{37}^{\left(1\right)}\\ D_{57}^{\left(1\right)}\\ D_{47}^{\left(1\right)}\end{array}\right)+r\left(\begin{array}{cc}{D}_{77}^{\left(1\right)}& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }\\ D_{57}^{\left(1\right)}& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}\\ D_{17}^{\left(1\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}\\ D_{37}^{\left(1\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}\\ D_{27}^{\left(1\right)}& D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)}\\ D_{47}^{\left(1\right)}& D_{45}^{\left(1\right)}-D_{45}^{\left(2\right)}\end{array}\right)-r\left(\begin{array}{cc}{D}_{77}^{\left(1\right)}& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }\\ D_{57}^{\left(1\right)}& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}\\ D_{17}^{\left(1\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}\\ D_{37}^{\left(1\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}\\ D_{27}^{\left(1\right)}& D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)}\end{array}\right)$$

$$+r\left(\begin{array}{ccc}{D}_{77}^{\left(1\right)}& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{37}^{\left(1\right)}\right)}_{\varphi }\\ D_{57}^{\left(1\right)}& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}& -{(D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)})}_{\varphi }\\ D_{37}^{\left(1\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}& D_{33}^{\left(1\right)}-D_{33}^{\left(2\right)}+D_{33}^{\left(3\right)}\\ D_{17}^{\left(1\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}& D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)}\\ D_{27}^{\left(1\right)}& D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)}& D_{23}^{\left(1\right)}-D_{23}^{\left(2\right)}+D_{23}^{\left(3\right)}\end{array}\right)-r\left(\begin{array}{ccc}{D}_{77}^{\left(1\right)}& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{37}^{\left(1\right)}\right)}_{\varphi }\\ D_{57}^{\left(1\right)}& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}& -{(D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)})}_{\varphi }\\ D_{37}^{\left(1\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}& D_{33}^{\left(1\right)}-D_{33}^{\left(2\right)}+D_{33}^{\left(3\right)}\\ D_{17}^{\left(1\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}& D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)}\end{array}\right)$$

$$\begin{array}{c}\hfill +{ln}_{\pm }\left(\begin{array}{cccc}{D}_{77}^{\left(1\right)}& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{37}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{17}^{\left(1\right)}\right)}_{\varphi }\\ D_{57}^{\left(1\right)}& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}& -{(D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)})}_{\varphi }& -{(D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)})}_{\varphi }\\ D_{37}^{\left(1\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}& D_{33}^{\left(1\right)}-D_{33}^{\left(2\right)}+D_{33}^{\left(3\right)}& -{(D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)})}_{\varphi }\\ D_{17}^{\left(1\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}& D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)}& D_{11}^{\left(1\right)}-D_{11}^{\left(2\right)}+D_{11}^{\left(3\right)}-D_{11}^{\left(4\right)}\end{array}\right).\end{array}$$

(20)

Step 4. The ranks and $\beta$($\varphi$)-signature of matrices in (4.5) and (4.6) can be represented by the following expression

$$r\left(\begin{array}{c}{\left(B_1\right)}_{\varphi }\\ C_1\end{array}\right)=r\left(\begin{array}{c}{B}_1\end{array}\right)+r\left(\begin{array}{c}{D}_{77}^{\left(1\right)}\\ D_{17}^{\left(1\right)}\\ D_{27}^{\left(1\right)}\\ D_{37}^{\left(1\right)}\\ D_{57}^{\left(1\right)}\\ D_{47}^{\left(1\right)}\\ D_{67}^{\left(1\right)}\end{array}\right),r\left(\begin{array}{cc}{A}_2& 0\\ 0& {\left(B_1\right)}_{\varphi }\\ B_1& C_1\end{array}\right)=r\left(\begin{array}{c}{A}_2\\ B_1\end{array}\right)+r\left(\begin{array}{c}{B}_1\end{array}\right)+r\left(\begin{array}{c}{D}_{77}^{\left(1\right)}\\ D_{17}^{\left(1\right)}\\ D_{27}^{\left(1\right)}\\ D_{37}^{\left(1\right)}\\ D_{57}^{\left(1\right)}\\ D_{47}^{\left(1\right)}\end{array}\right),$$

$$r\left(\begin{array}{ccc}{\left(B_2\right)}_{\varphi }& 0& 0\\ -C_2& A_2& 0\\ {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& B_1& C_1\end{array}\right)=r\left(\begin{array}{c}{A}_2\\ B_1\end{array}\right)+r\left(\begin{array}{cc}{B}_2& A_2\\ 0& B_1\end{array}\right)+r\left(\begin{array}{cc}{D}_{77}^{\left(1\right)}& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }\\ D_{57}^{\left(1\right)}& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}\\ D_{17}^{\left(1\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}\\ D_{37}^{\left(1\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}\\ D_{27}^{\left(1\right)}& D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)}\\ D_{47}^{\left(1\right)}& D_{45}^{\left(1\right)}-D_{45}^{\left(2\right)}\end{array}\right),$$

$$r\left(\begin{array}{cccc}{A}_3& 0& 0& 0\\ 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ B_2& -C_2& A_2& 0\\ 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& B_1& C_1\end{array}\right)=r\left(\begin{array}{cc}{A}_3& 0\\ B_2& A_2\\ 0& B_1\end{array}\right)+r\left(\begin{array}{cc}{B}_2& A_2\\ 0& B_1\end{array}\right)+r\left(\begin{array}{cc}{D}_{77}^{\left(1\right)}& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }\\ D_{57}^{\left(1\right)}& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}\\ D_{17}^{\left(1\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}\\ D_{37}^{\left(1\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}\\ D_{27}^{\left(1\right)}& D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)}\end{array}\right),$$

$$r\left(\begin{array}{ccccc}{\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ C_3& A_3& 0& 0& 0\\ {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& B_2& -C_2& A_2& 0\\ 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& B_1& C_1\end{array}\right)=r\left(\begin{array}{cc}{A}_3& 0\\ B_2& A_2\\ 0& B_1\end{array}\right)+r\left(\begin{array}{ccc}{B}_3& A_3& 0\\ 0& B_2& A_2\\ 0& 0& B_1\end{array}\right)+r\left(\begin{array}{ccc}{D}_{77}^{\left(1\right)}& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{37}^{\left(1\right)}\right)}_{\varphi }\\ D_{57}^{\left(1\right)}& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}& -{(D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)})}_{\varphi }\\ D_{37}^{\left(1\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}& D_{33}^{\left(1\right)}-D_{33}^{\left(2\right)}+D_{33}^{\left(3\right)}\\ D_{17}^{\left(1\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}& D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)}\\ D_{27}^{\left(1\right)}& D_{25}^{\left(1\right)}-D_{25}^{\left(2\right)}& D_{23}^{\left(1\right)}-D_{23}^{\left(2\right)}+D_{23}^{\left(3\right)}\end{array}\right),$$

$$r\left(\begin{array}{cccccc}{A}_4& 0& 0& 0& 0& 0\\ 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ B_3& C_3& A_3& 0& 0& 0\\ 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& B_1& C_1\end{array}\right)$$

$$=r\left(\begin{array}{ccc}{B}_3& A_3& 0\\ 0& B_2& A_2\\ 0& 0& B_1\end{array}\right)+r\left(\begin{array}{ccc}{A}_4& 0& 0\\ B_3& A_3& 0\\ 0& B_2& A_2\\ 0& 0& B_1\end{array}\right)+r\left(\begin{array}{ccc}{D}_{77}^{\left(1\right)}& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{37}^{\left(1\right)}\right)}_{\varphi }\\ D_{57}^{\left(1\right)}& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}& -{(D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)})}_{\varphi }\\ D_{37}^{\left(1\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}& D_{33}^{\left(1\right)}-D_{33}^{\left(2\right)}+D_{33}^{\left(3\right)}\\ D_{17}^{\left(1\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}& D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)}\end{array}\right),$$

$${ln}_{\pm }\left(\begin{array}{ccccccc}-C_4& A_4& 0& 0& 0& 0& 0\\ {\left(A_4\right)}_{\varphi }& 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ 0& B_3& C_3& A_3& 0& 0& 0\\ 0& 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& 0& B_1& C_1\end{array}\right)$$

$$=r\left(\begin{array}{ccc}{A}_4& 0& 0\\ B_3& A_3& 0\\ 0& B_2& A_2\\ 0& 0& B_1\end{array}\right)+{ln}_{\pm }\left(\begin{array}{cccc}{D}_{77}^{\left(1\right)}& -{\left(D_{57}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{37}^{\left(1\right)}\right)}_{\varphi }& -{\left(D_{17}^{\left(1\right)}\right)}_{\varphi }\\ D_{57}^{\left(1\right)}& D_{55}^{\left(1\right)}-D_{55}^{\left(2\right)}& -{(D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)})}_{\varphi }& -{(D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)})}_{\varphi }\\ D_{37}^{\left(1\right)}& D_{35}^{\left(1\right)}-D_{35}^{\left(2\right)}& D_{33}^{\left(1\right)}-D_{33}^{\left(2\right)}+D_{33}^{\left(3\right)}& -{(D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)})}_{\varphi }\\ D_{17}^{\left(1\right)}& D_{15}^{\left(1\right)}-D_{15}^{\left(2\right)}& D_{13}^{\left(1\right)}-D_{13}^{\left(2\right)}+D_{13}^{\left(3\right)}& D_{11}^{\left(1\right)}-D_{11}^{\left(2\right)}+D_{11}^{\left(3\right)}-D_{11}^{\left(4\right)}\end{array}\right),$$

$$n_2=r\left(\begin{array}{cc}{B}_1& 0\\ A_2& B_2\end{array}\right)+r\left(\begin{array}{cccc}{A}_1& B_1& 0& 0\\ 0& A_2& B_2& 0\\ 0& 0& A_3& B_3\end{array}\right)-r\left(\begin{array}{ccc}{A}_1& B_1& 0\\ 0& A_2& B_2\end{array}\right)-r\left(\begin{array}{ccc}{B}_1& 0& 0\\ A_2& B_2& 0\\ 0& A_3& B_3\end{array}\right),$$

$$n_4=r\left(\begin{array}{c}{B}_1\end{array}\right)+r\left(\begin{array}{ccc}{A}_1& B_1& 0\\ 0& A_2& B_2\end{array}\right)-r\left(\begin{array}{cc}{A}_1& B_1\end{array}\right)-r\left(\begin{array}{cc}{B}_1& 0\\ A_2& B_2\end{array}\right),$$

$$n_6=r\left(\begin{array}{cc}{A}_1& B_1\end{array}\right)-r\left(\begin{array}{c}{B}_1\end{array}\right),r_{a_1}=r\left(\begin{array}{c}{A}_1\end{array}\right).$$

The above results indicate that the $\beta$($\varphi$)-signature bounds of the $\varphi$-skew-Hermitian matrix ${\widehat{X}}_1$ can be expressed as

$$max{ln}_{\pm }\left({\widehat{X}}_1\right)=t_1-r\left(\begin{array}{c}{A}_1\end{array}\right)-r\left(\begin{array}{ccc}{B}_1& 0& 0\\ A_2& B_2& 0\\ 0& A_3& B_3\end{array}\right)-r\left(\begin{array}{ccc}{A}_4& 0& 0\\ B_3& A_3& 0\\ 0& B_2& A_2\\ 0& 0& B_1\end{array}\right)$$

$$+r\left(\begin{array}{cccc}{A}_1& B_1& 0& 0\\ 0& A_2& B_2& 0\\ 0& 0& A_3& B_3\end{array}\right)+{ln}_{\pm }\left(\begin{array}{ccccccc}-C_4& A_4& 0& 0& 0& 0& 0\\ {\left(A_4\right)}_{\varphi }& 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ 0& B_3& C_3& A_3& 0& 0& 0\\ 0& 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& 0& B_1& C_1\end{array}\right),$$

$$min{ln}_{\pm }\left({\widehat{X}}_1\right)=r\left(\begin{array}{c}{\left(B_1\right)}_{\varphi }\\ C_1\end{array}\right)-r\left(\begin{array}{cc}{A}_2& 0\\ 0& {\left(B_1\right)}_{\varphi }\\ B_1& C_1\end{array}\right)+r\left(\begin{array}{ccc}{\left(B_2\right)}_{\varphi }& 0& 0\\ -C_2& A_2& 0\\ {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& B_1& C_1\end{array}\right)$$

$$-r\left(\begin{array}{cccc}{A}_3& 0& 0& 0\\ 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ B_2& -C_2& A_2& 0\\ 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& B_1& C_1\end{array}\right)+r\left(\begin{array}{ccccc}{\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ C_3& A_3& 0& 0& 0\\ {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& B_2& -C_2& A_2& 0\\ 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& B_1& C_1\end{array}\right)$$

$$-r\left(\begin{array}{cccccc}{A}_4& 0& 0& 0& 0& 0\\ 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ B_3& C_3& A_3& 0& 0& 0\\ 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& B_1& C_1\end{array}\right)+{ln}_{\pm }\left(\begin{array}{ccccccc}-C_4& A_4& 0& 0& 0& 0& 0\\ {\left(A_4\right)}_{\varphi }& 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ 0& B_3& C_3& A_3& 0& 0& 0\\ 0& 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& 0& B_1& C_1\end{array}\right).$$

In conclusion, Theorem 2 is proved. □

Based on Theorem 2, we derive some necessary and sufficient conditions for the system (1) to have $\beta$($\varphi$)-positive definite, $\beta$($\varphi$)-positive semidefinite, $\beta$($\varphi$)-negative definite and $\beta$($\varphi$)-negative semidefinite solutions.

Theorem 3.

Assume that the system (1) has a ϕ-skew-Hermitian solution $(X_1,X_2,X_3,X_4)\in {\mathbb{H}}^{t_1\times t_1}\times {\mathbb{H}}^{t_2\times t_2}\times {\mathbb{H}}^{t_3\times t_3}\times {\mathbb{H}}^{t_4\times t_4}$. We can derive the following conclusions.

$\left(a\right)$ There is a β(ϕ)-positive definite solution $X_1$ if and only if

$${ln}_{+}\left(\begin{array}{ccccccc}-C_4& A_4& 0& 0& 0& 0& 0\\ {\left(A_4\right)}_{\varphi }& 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ 0& B_3& C_3& A_3& 0& 0& 0\\ 0& 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& 0& B_1& C_1\end{array}\right)$$

$$=r\left(\begin{array}{c}{A}_1\end{array}\right)+r\left(\begin{array}{ccc}{B}_1& 0& 0\\ A_2& B_2& 0\\ 0& A_3& B_3\end{array}\right)+r\left(\begin{array}{ccc}{A}_4& 0& 0\\ B_3& A_3& 0\\ 0& B_2& A_2\\ 0& 0& B_1\end{array}\right)-r\left(\begin{array}{cccc}{A}_1& B_1& 0& 0\\ 0& A_2& B_2& 0\\ 0& 0& A_3& B_3\end{array}\right).$$

$\left(b\right)$ There is a β(ϕ)-negative definite solution $X_1$ if and only if

$${ln}_{-}\left(\begin{array}{ccccccc}-C_4& A_4& 0& 0& 0& 0& 0\\ {\left(A_4\right)}_{\varphi }& 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ 0& B_3& C_3& A_3& 0& 0& 0\\ 0& 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& 0& B_1& C_1\end{array}\right)$$

$$=r\left(\begin{array}{c}{A}_1\end{array}\right)+r\left(\begin{array}{ccc}{B}_1& 0& 0\\ A_2& B_2& 0\\ 0& A_3& B_3\end{array}\right)+r\left(\begin{array}{ccc}{A}_4& 0& 0\\ B_3& A_3& 0\\ 0& B_2& A_2\\ 0& 0& B_1\end{array}\right)-r\left(\begin{array}{cccc}{A}_1& B_1& 0& 0\\ 0& A_2& B_2& 0\\ 0& 0& A_3& B_3\end{array}\right).$$

$\left(c\right)$ There is a β(ϕ)-positive semidefinite solution $X_1$ if and only if

$${ln}_{+}\left(\begin{array}{ccccccc}-C_4& A_4& 0& 0& 0& 0& 0\\ {\left(A_4\right)}_{\varphi }& 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ 0& B_3& C_3& A_3& 0& 0& 0\\ 0& 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& 0& B_1& C_1\end{array}\right)$$

$$=r\left(\begin{array}{cc}{A}_2& 0\\ 0& {\left(B_1\right)}_{\varphi }\\ B_1& C_1\end{array}\right)+r\left(\begin{array}{cccc}{A}_3& 0& 0& 0\\ 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ B_2& -C_2& A_2& 0\\ 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& B_1& C_1\end{array}\right)+r\left(\begin{array}{cccccc}{A}_4& 0& 0& 0& 0& 0\\ 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ B_3& C_3& A_3& 0& 0& 0\\ 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& B_1& C_1\end{array}\right)$$

$$-r\left(\begin{array}{c}{\left(B_1\right)}_{\varphi }\\ C_1\end{array}\right)-r\left(\begin{array}{ccc}{\left(B_2\right)}_{\varphi }& 0& 0\\ -C_2& A_2& 0\\ {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& B_1& C_1\end{array}\right)-r\left(\begin{array}{ccccc}{\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ C_3& A_3& 0& 0& 0\\ {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& B_2& -C_2& A_2& 0\\ 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& B_1& C_1\end{array}\right).$$

$\left(d\right)$ There is a β(ϕ)-negative semidefinite solution $X_1$ if and only if

$${ln}_{-}\left(\begin{array}{ccccccc}-C_4& A_4& 0& 0& 0& 0& 0\\ {\left(A_4\right)}_{\varphi }& 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ 0& B_3& C_3& A_3& 0& 0& 0\\ 0& 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& 0& B_1& C_1\end{array}\right)$$

$$=r\left(\begin{array}{cc}{A}_2& 0\\ 0& {\left(B_1\right)}_{\varphi }\\ B_1& C_1\end{array}\right)+r\left(\begin{array}{cccc}{A}_3& 0& 0& 0\\ 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ B_2& -C_2& A_2& 0\\ 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& B_1& C_1\end{array}\right)+r\left(\begin{array}{cccccc}{A}_4& 0& 0& 0& 0& 0\\ 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ B_3& C_3& A_3& 0& 0& 0\\ 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& B_1& C_1\end{array}\right)$$

$$-r\left(\begin{array}{c}{\left(B_1\right)}_{\varphi }\\ C_1\end{array}\right)-r\left(\begin{array}{ccc}{\left(B_2\right)}_{\varphi }& 0& 0\\ -C_2& A_2& 0\\ {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& B_1& C_1\end{array}\right)-r\left(\begin{array}{ccccc}{\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ C_3& A_3& 0& 0& 0\\ {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& B_2& -C_2& A_2& 0\\ 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& B_1& C_1\end{array}\right).$$

$\left(e\right)$ All the solutions $X_1$ are β(ϕ)-positive definite if and only if

$$r\left(\begin{array}{c}{\left(B_1\right)}_{\varphi }\\ C_1\end{array}\right)-r\left(\begin{array}{cc}{A}_2& 0\\ 0& {\left(B_1\right)}_{\varphi }\\ B_1& C_1\end{array}\right)+r\left(\begin{array}{ccc}{\left(B_2\right)}_{\varphi }& 0& 0\\ -C_2& A_2& 0\\ {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& B_1& C_1\end{array}\right)$$

$$-r\left(\begin{array}{cccc}{A}_3& 0& 0& 0\\ 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ B_2& -C_2& A_2& 0\\ 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& B_1& C_1\end{array}\right)+r\left(\begin{array}{ccccc}{\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ C_3& A_3& 0& 0& 0\\ {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& B_2& -C_2& A_2& 0\\ 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& B_1& C_1\end{array}\right)$$

$$-r\left(\begin{array}{cccccc}{A}_4& 0& 0& 0& 0& 0\\ 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ B_3& C_3& A_3& 0& 0& 0\\ 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& B_1& C_1\end{array}\right)+{ln}_{-}\left(\begin{array}{ccccccc}-C_4& A_4& 0& 0& 0& 0& 0\\ {\left(A_4\right)}_{\varphi }& 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ 0& B_3& C_3& A_3& 0& 0& 0\\ 0& 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& 0& B_1& C_1\end{array}\right)=t_1.$$

$\left(f\right)$ All the solutions $X_1$ are β(ϕ)-negative definite if and only if

$$r\left(\begin{array}{c}{\left(B_1\right)}_{\varphi }\\ C_1\end{array}\right)-r\left(\begin{array}{cc}{A}_2& 0\\ 0& {\left(B_1\right)}_{\varphi }\\ B_1& C_1\end{array}\right)+r\left(\begin{array}{ccc}{\left(B_2\right)}_{\varphi }& 0& 0\\ -C_2& A_2& 0\\ {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& B_1& C_1\end{array}\right)$$

$$-r\left(\begin{array}{cccc}{A}_3& 0& 0& 0\\ 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ B_2& -C_2& A_2& 0\\ 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& B_1& C_1\end{array}\right)+r\left(\begin{array}{ccccc}{\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ C_3& A_3& 0& 0& 0\\ {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& B_2& -C_2& A_2& 0\\ 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& B_1& C_1\end{array}\right)$$

$$-r\left(\begin{array}{cccccc}{A}_4& 0& 0& 0& 0& 0\\ 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ B_3& C_3& A_3& 0& 0& 0\\ 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& B_1& C_1\end{array}\right)+{ln}_{+}\left(\begin{array}{ccccccc}-C_4& A_4& 0& 0& 0& 0& 0\\ {\left(A_4\right)}_{\varphi }& 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ 0& B_3& C_3& A_3& 0& 0& 0\\ 0& 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& 0& B_1& C_1\end{array}\right)=t_1.$$

$\left(g\right)$ All the solutions $X_1$ are β(ϕ)-positive semidefinite if and only if

$$t_1+{ln}_{-}\left(\begin{array}{ccccccc}-C_4& A_4& 0& 0& 0& 0& 0\\ {\left(A_4\right)}_{\varphi }& 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ 0& B_3& C_3& A_3& 0& 0& 0\\ 0& 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& 0& B_1& C_1\end{array}\right)$$

$$=r\left(\begin{array}{c}{A}_1\end{array}\right)+r\left(\begin{array}{ccc}{B}_1& 0& 0\\ A_2& B_2& 0\\ 0& A_3& B_3\end{array}\right)+r\left(\begin{array}{ccc}{A}_4& 0& 0\\ B_3& A_3& 0\\ 0& B_2& A_2\\ 0& 0& B_1\end{array}\right)-r\left(\begin{array}{cccc}{A}_1& B_1& 0& 0\\ 0& A_2& B_2& 0\\ 0& 0& A_3& B_3\end{array}\right).$$

$\left(h\right)$ All the solutions $X_1$ are β(ϕ)-negative semidefinite if and only if

$$t_1+{ln}_{+}\left(\begin{array}{ccccccc}-C_4& A_4& 0& 0& 0& 0& 0\\ {\left(A_4\right)}_{\varphi }& 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ 0& B_3& C_3& A_3& 0& 0& 0\\ 0& 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& 0& B_1& C_1\end{array}\right)$$

$$=r\left(\begin{array}{c}{A}_1\end{array}\right)+r\left(\begin{array}{ccc}{B}_1& 0& 0\\ A_2& B_2& 0\\ 0& A_3& B_3\end{array}\right)+r\left(\begin{array}{ccc}{A}_4& 0& 0\\ B_3& A_3& 0\\ 0& B_2& A_2\\ 0& 0& B_1\end{array}\right)-r\left(\begin{array}{cccc}{A}_1& B_1& 0& 0\\ 0& A_2& B_2& 0\\ 0& 0& A_3& B_3\end{array}\right).$$

Proof. 

According to Theorem 2, the system (1) has a $\beta$($\varphi$)-positive definite solution $X_1$ if and only if

$$max{ln}_{+}\left(X_1\right)=t_1.$$

It follows from Theorem 2 that

$$\begin{array}{c}\hfill max{ln}_{\pm }\left(X_1\right)=t_1-r\left(\begin{array}{c}{A}_1\end{array}\right)-r\left(\begin{array}{ccc}{B}_1& 0& 0\\ A_2& B_2& 0\\ 0& A_3& B_3\end{array}\right)-r\left(\begin{array}{ccc}{A}_4& 0& 0\\ B_3& A_3& 0\\ 0& B_2& A_2\\ 0& 0& B_1\end{array}\right)\end{array}$$

$$\begin{array}{c}\hfill +r\left(\begin{array}{cccc}{A}_1& B_1& 0& 0\\ 0& A_2& B_2& 0\\ 0& 0& A_3& B_3\end{array}\right)+{ln}_{\pm }\left(\begin{array}{ccccccc}-C_4& A_4& 0& 0& 0& 0& 0\\ {\left(A_4\right)}_{\varphi }& 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ 0& B_3& C_3& A_3& 0& 0& 0\\ 0& 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& 0& B_1& C_1\end{array}\right)=t_1.\end{array}$$

Then, we can obtain that there is a $\beta$($\varphi$)-positive definite solution $X_1$ if and only if

$${ln}_{+}\left(\begin{array}{ccccccc}-C_4& A_4& 0& 0& 0& 0& 0\\ {\left(A_4\right)}_{\varphi }& 0& {\left(B_3\right)}_{\varphi }& 0& 0& 0& 0\\ 0& B_3& C_3& A_3& 0& 0& 0\\ 0& 0& {\left(A_3\right)}_{\varphi }& 0& {\left(B_2\right)}_{\varphi }& 0& 0\\ 0& 0& 0& B_2& -C_2& A_2& 0\\ 0& 0& 0& 0& {\left(A_2\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ 0& 0& 0& 0& 0& B_1& C_1\end{array}\right)$$

$$=r\left(\begin{array}{c}{A}_1\end{array}\right)+r\left(\begin{array}{ccc}{B}_1& 0& 0\\ A_2& B_2& 0\\ 0& A_3& B_3\end{array}\right)+r\left(\begin{array}{ccc}{A}_4& 0& 0\\ B_3& A_3& 0\\ 0& B_2& A_2\\ 0& 0& B_1\end{array}\right)-r\left(\begin{array}{cccc}{A}_1& B_1& 0& 0\\ 0& A_2& B_2& 0\\ 0& 0& A_3& B_3\end{array}\right).$$

Hence, we can prove the statements (a). In a similar way, we can obtain (b)–(h). □

5. Conclusions

We have provided the general solution to the system (1). Furthermore, we have given the $\beta$($\varphi$)-signature bounds of the $\varphi$-skew-Hermitian solution to the system (1). Finally, we have presented some necessary and sufficient conditions for the system (1) to have $\beta$($\varphi$)-positive definite, $\beta$($\varphi$)-positive semidefinite, $\beta$($\varphi$)-negative definite and $\beta$($\varphi$)-negative semidefinite solutions.