Solving the QLY Least Squares Problem of Dual Quaternion Matrix Equation Based on STP of Dual Quaternion Matrices

Abstract

Dual algebra plays an important role in kinematic synthesis and dynamic analysis, but there are still few studies on dual quaternion matrix theory. This paper provides an efficient method for solving the QLY least squares problem of the dual quaternion matrix equation $AXB+CYD\approx E$, where X, Y are unknown dual quaternion matrices with special structures. First, we define a semi-tensor product of dual quaternion matrices and study its properties, which can be used to achieve the equivalent form of the dual quaternion matrix equation. Then, by using the dual representation of dual quaternion and the $\mathcal{GH}$-representation of special dual quaternion matrices, we study the expression of QLY least squares Hermitian solution of the dual quaternion matrix equation $AXB+CYD\approx E$. The algorithm is given and the numerical examples are provided to illustrate the efficiency of the method.

1. Introduction

In this paper, $\mathbb{R}/\mathbb{Q}/\mathbb{D}/\mathbb{DQ}/{\mathbb{R}}_n/{\mathbb{R}}^n/{\mathbb{D}}_n/{\mathbb{D}}^n$ denote the sets of all real numbers, quaternions, dual numbers, dual quaternions, real row vectors, real column vectors, dual row vectors, and dual column vectors with n-dimension, respectively. ${\mathbb{R}}^{m\times n}$/${\mathbb{Q}}^{m\times n}/$${\mathbb{D}}^{m\times n}$/${\mathbb{DQ}}^{m\times n}$ denote the sets of all $m\times n$ real matrices, quaternion matrices, dual matrices, and dual quaternion matrices, respectively. ${\mathbb{R}}_S^{n\times n}$/${\mathbb{R}}_A^{n\times n}$/${\mathbb{DQ}}_H^{n\times n}$ denote the sets of all $n\times n$ real symmetric matrices, real anti-symmetric matrices, and dual quaternion Hermitian matrices, respectively. ${\mathbb{O}}^n/$${\mathbb{O}}^{m\times n}$ denote the n-dimensional zero vector and $m\times n$ zero matrix, respectively. $A^H$ and $A^{†}$ represent the conjugate transpose matrix and the Moore–Penrose generalized inverse matrix of A, respectively. ${\delta }_n^i$ represents the i-th column of identity matrix $I_n$. $A\otimes B$ represents the Kronecker product of matrices. $\parallel ·\parallel$ represents the Frobenius norm of a quaternion matrix or Euclidean norm of a quaternion vector, ${\parallel ·\parallel }_F$ represents the Frobenius norm of a dual quaternion matrix. ${\parallel ·\parallel }_{2\left(DQ\right)}$ represents the 2-norm of a dual quaternion vector. ${\parallel ·\parallel }_{2\left(D\right)}$ represents the 2-norm of a dual vector.

As a powerful tool, dual algebra has often been used in kinematic synthesis and dynamic analysis of spatial mechanisms to search the closed form solutions. In [1], Yang et al. obtained the dynamic equations for an offset unsymmetric gyroscope with an obliquely placed rotor by using $(3\times 3)$ matrices of dual number elements. In [2], Veldkamp introduced the application to theoretical space kinematics of the algebra of dual quantities and the differential-geometry of dual curves. In [3], Angeles et al. revisited dual algebra in the context of kinematic analysis, which led to an introduction of dual quaternions, and proved that the calculation of the finite screw and the instant screw of a rigid body is reduced to the computational problem of correlation dual matrices. Among them, the application of dual quaternion matrices in kinematic synthesis and dynamic analysis of spatial mechanisms is a significant research topic.

Dual quaternion is an important part of dual algebra, which is used to study non-Euclidean geometry. Nowadays, dual quaternions have wide applications in kinematic analysis, automatic differentiation, rigid body motion analysis and robotics [4,5,6,7,8,9]. At the same time, scholars have also made great achievements in dual quaternion theory. Qi et al. [10] introduced right and left eigenvalues for the square dual quaternion matrix and presented the unitary decomposition of a dual quaternion Hermitian matrix. Ling et al. [11] studied some related basic properties of a dual quaternion matrix and introduced the concept of generalized inverses of dual quaternion matrices. Cui et al. [12] provided a power method to compute the dominant eigenvalue of a dual quaternion Hermitian matrix.

As one of the research contents of Clifford algebra, linear matrix equations are widely used in various fields and attract many scholars to study them [13,14,15,16,17,18,19], including the study of dual matrix equations and dual quaternion matrix equations. For example, Chen et al. [20] used the rank and Moore–Penrose inverses of matrices to give the solvability conditions for the dual quaternion matrix equation $AXB=C$ and derived the expression of its general solution. Zeng et al. [21] used the singular value decomposition of quaternion matrices to present the expression of the general solution of the dual quaternion matrix equation $A^{T}X+X^{T}A=D$. In [22], Qi, Ling and Yan defined a total order for dual numbers. Wang et al. [23] expanded it to dual vectors and introduced a QLY total order for dual vectors. The specific definition is as follows. Consider two dual vectors $m=m_{st}+m_Iϵ$, $n=n_{st}+n_Iϵ$. We have $m\stackrel{Q}{<}n$ if $\parallel m_{st}\parallel <\parallel n_{st}\parallel$ or $\parallel m_{st}\parallel =\parallel n_{st}\parallel$ and $\parallel m_I\parallel <\parallel n_I\parallel$. We have $m\stackrel{Q}{=}n$ if and only if $\parallel m_{st}\parallel =\parallel n_{st}\parallel$ and $\parallel m_I\parallel =\parallel n_I\parallel .$ Based on this, they considered the QLY minimal norm least squares problem of the dual matrix equation and provided its compact formula, that is, for the inconsistent dual equation $Ax\approx b$, where $A=A_{st}+A_Iϵ\in {\mathbb{D}}^{m\times n}$ and $b=b_{st}+b_Iϵ\in {\mathbb{D}}^m$, if there exists $x_Q=x_{st}+x_Iϵ\in {\mathbb{D}}^n$ such that

$$\left\{\begin{array}{c}\parallel A_{st}{x}_{st}-b_{st}\parallel =min,\hfill \\ \parallel A_{st}{x}_I+A_I{x}_{st}-b_I\parallel =min,\hfill \end{array}\right.$$

then $x_Q$ is a QLY least squares solution to the dual equation.

In this paper, we transfer Wang’s thought of solving the QLY least squares problem of dual matrix equation $Ax\approx b$ to solving the least squares dual quaternion matrix equation and call it the QLY least squares problem of the dual quaternion matrix equation. By using the semi-tensor product (STP) of dual quaternion matrices, we study the QLY least squares Hermitian solution of the dual quaternion matrix equation

$$AXB+CYD\approx E.$$

(1)

The specific problem in this paper is as follows:

Problem 1.

Let $A\in {\mathbb{DQ}}^{m\times n}$, $B\in {\mathbb{DQ}}^{n\times p}$, $C\in {\mathbb{DQ}}^{m\times q}$, $D\in {\mathbb{DQ}}^{q\times p}$, $E\in {\mathbb{DQ}}^{m\times p}$, and $X\in {\mathbb{DQ}}_H^{n\times n}$, $Y\in {\mathbb{DQ}}_H^{q\times q}$ are unknown dual quaternion Hermitian matrices. Denote

$$J_Q=\left\{(X,Y)|X\in {\mathbb{DQ}}_H^{n\times n},Y\in {\mathbb{DQ}}_H^{q\times q},{\parallel AXB+CYD-E\parallel }_F=mi{n}_Q\right\},$$

where $mi{n}_Q$ is

$$\left\{\begin{array}{c}\parallel A_{st}{X}_{st}{B}_{st}+C_{st}{Y}_{st}{D}_{st}-E_{st}\parallel =min,\hfill \\ \parallel A_{st}{X}_{st}{B}_I+A_{st}{X}_I{B}_{st}+A_I{X}_{st}{B}_{st}+C_{st}{Y}_{st}{D}_I+C_{st}{Y}_I{D}_{st}+C_I{Y}_{st}{D}_{st}-E_I\parallel =min.\hfill \end{array}\right.$$

Find out $(X_H,Y_H)\in J_Q$ such that

$$\parallel (X_H,Y_H){\parallel }_F=mi{n}_Q{|}_{(X,Y)\in J_Q}{\parallel (X,Y)\parallel }_F.$$

$(X_H,Y_H)$ in Problem 1 is called the QLY minimal norm least squares Hermitian solution of (1).

This paper is organized as follows. In Section 2, we first introduce the relevant knowledge of dual numbers, dual quaternions, dual (quaternion) vectors and dual (quaternion) matrices, and provide the dual representation of a dual quaternion matrix. Then, we define the semi-tensor product (STP) of dual quaternion matrices and propose the results of a vector operator on dual quaternion matrices. In Section 3, we present the $\mathcal{GH}$-representation of special dual quaternion matrices and use the dual representation of dual quaternion matrices and the related properties of the vector operator to transform the QLY least squares of the linear dual quaternion least squares problem into the QLY least squares of the linear dual least squares problem and derive the QLY least squares Hermitian solution of dual quaternion matrix Equation (1). In Section 4, we provide the numerical algorithm and the numerical examples for demonstrating the efficiency of the method. Finally, in Section 5, we provide some concluding remarks.

2. Preliminaries

2.1. The Relevant Knowledge of Dual Vectors (Matrices) and Dual Quaternion Vectors (Matrices)

First, we will introduce the relevant knowledge [11,12,22] of dual numbers (quaternions), dual (quaternion) vectors and matrices.

The set of dual numbers and dual quaternions are denoted by

$$\mathbb{D}=\{d=d_{st}+d_Iϵ|d_{st},d_I\in \mathbb{R},ϵ\ne 0,{ϵ}^2=0\},$$

$$\mathbb{DQ}=\{q=q_{st}+q_Iϵ|q_{st},q_I\in \mathbb{Q},ϵ\ne 0,{ϵ}^2=0\}.$$

We call $d_{st}/q_{st}$ the real part or the standard part of $d/q$, and $d_I/q_I$ the dual part or the infinitesimal part of $d/q$. It is worth noting that $ϵ$ is the infinitesimal unit, which is commutative in multiplication with real numbers and quaternions.

For any two dual quaternions $a=a_{st}+a_Iϵ$, $b=b_{st}+b_Iϵ\in \mathbb{D}/\mathbb{DQ}$, we have

$$a+b=(a_{st}+b_{st})+(a_I+b_I)ϵ,$$

$$ab=a_{st}{b}_{st}+(a_{st}{b}_I+a_I{b}_{st})ϵ.$$

For example, let $a=3+2ϵ$, $b=1+3ϵ$, then

$$a+b=(3+2ϵ)+(1+3ϵ)=4+5ϵ,$$

$$ab=(3+2ϵ)(1+3ϵ)=3+(9+2)ϵ+6{ϵ}^2=3+11ϵ.$$

If $d>0$, we say d is a positive dual number; and if $d\ge 0$, we say that d is a nonnegative dual number. If $d_{st}\ne 0$, we say that d is appreciable; otherwise, we say that d is infinitesimal.

For given dual number $d=d_{st}+d_Iϵ$, the magnitude is

$$\left|d\right|=\left\{\begin{array}{c}|d_{st}|+sgn\left(d_{st}\right)d_Iϵ,d_{st}\ne 0,\hfill \\ |d_I|ϵ,d_{st}=0,\hfill \end{array}\right.$$

where for any $u\in \mathbb{R}$,

$$sgn\left(u\right)=\left\{\begin{array}{c}1,ifu>0,\hfill \\ 0,ifu=0,\hfill \\ -1,ifu<0.\hfill \end{array}\right.$$

If d is nonnegative and appreciable, then the square root of d is still a nonnegative dual number. If d is positive and appreciable, we have

$$\sqrt{d}=\sqrt{d_{st}}+{\displaystyle \frac{d_I}{2\sqrt{d_{st}}}}ϵ.$$

When $d=0$, we have $\sqrt{d}=0.$

The conjugate of a dual quaternion q is defined by $\overline{q}=\overline{q_{st}}+\overline{q_I}ϵ$, and the magnitude of q is

$$|q|=\left\{\begin{array}{c}|q_{st}|+{\displaystyle \frac{q_{st}\overline{q_I}+q_I\overline{q_{st}}}{2|q_{st}|}}ϵ,if{q}_{st}\ne 0,\hfill \\ |q_I|ϵ,otherwise.\hfill \end{array}\right.$$

The set of n-dimensional dual quaternion vectors and $m\times n$ dual quaternion matrices are denoted by

$${\mathbb{DQ}}^n=\{x=x_{st}+x_Iϵ|x_{st},x_I\in {\mathbb{Q}}^n\},$$

$${\mathbb{DQ}}^{m\times n}=\{A=A_{st}+A_Iϵ|A_{st},A_I\in {\mathbb{Q}}^{m\times n}\}.$$

The 2-norm of dual quaternion vector $x\in {\mathbb{DQ}}^n$ is

$${\parallel x\parallel }_{2\left(DQ\right)}=\left\{\begin{array}{c}\sqrt{{\displaystyle \sum _{i=1}^n}{\left|x_i\right|}^2},if{x}_{st}\ne {\mathbb{O}}^n,\hfill \\ \sqrt{{\displaystyle \sum _{i=1}^n}{\left|{\left(x_i\right)}_I\right|}^2}ϵ,if{x}_{st}={\mathbb{O}}^n.\hfill \end{array}\right.$$

The F-norm of dual quaternion matrix $A=\left(a_{ij}\right)\in {\mathbb{DQ}}^{m\times n}$ is

$${\parallel A\parallel }_F=\left\{\begin{array}{c}\sqrt{{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\left|a_{ij}\right|}^2},if{A}_{st}\ne {\mathbb{O}}^{m\times n},\hfill \\ \sqrt{{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\left|{\left(a_{ij}\right)}_I\right|}^2}ϵ,if{A}_{st}={\mathbb{O}}^{m\times n}.\hfill \end{array}\right.$$

The conjugate transpose of A is denoted by

$$A^H=A_{st}^H+A_I^Hϵ.$$

A is called a dual quaternion Hermitian matrix if $A^H=A.$ It is easy to know that the real parts of the standard part and the dual part are real symmetric matrices, and the imaginary parts are real anti-symmetric matrices.

The dual matrix representation of the dual quaternion matrix is shown in [24], which is used for equivalent transformation in solving the QLY least squares problem of dual quaternion matrix Equation (1).

Definition 1.

A dual quaternion matrix $A\in {\mathbb{DQ}}^{m\times n}$ can be uniquely represented as

$$A=({\left(A_1\right)}_{st}+{\left(A_1\right)}_Iϵ)+({\left(A_2\right)}_{st}+{\left(A_2\right)}_Iϵ)i+({\left(A_3\right)}_{st}+{\left(A_3\right)}_Iϵ)j+({\left(A_4\right)}_{st}+{\left(A_4\right)}_Iϵ)k,$$

where ${\left(A_m\right)}_{st}$, ${\left(A_m\right)}_I\in {\mathbb{R}}^{m\times n}$, $m=1,2,3,4$. The dual matrix representation of dual quaternion matrix A is

$$\begin{array}{c}\hfill \psi \left(A\right)=\left[\begin{array}{cccc}{\left(A_1\right)}_{st}+{\left(A_1\right)}_Iϵ& -{\left(A_2\right)}_{st}-{\left(A_2\right)}_Iϵ& -{\left(A_3\right)}_{st}-{\left(A_3\right)}_Iϵ& -{\left(A_4\right)}_{st}-{\left(A_4\right)}_Iϵ\\ {\left(A_2\right)}_{st}+{\left(A_2\right)}_Iϵ& {\left(A_1\right)}_{st}+{\left(A_1\right)}_Iϵ& -{\left(A_4\right)}_{st}-{\left(A_4\right)}_Iϵ& {\left(A_3\right)}_{st}+{\left(A_3\right)}_Iϵ\\ {\left(A_3\right)}_{st}+{\left(A_3\right)}_Iϵ& {\left(A_4\right)}_{st}+{\left(A_4\right)}_Iϵ& {\left(A_1\right)}_{st}+{\left(A_1\right)}_Iϵ& -{\left(A_2\right)}_{st}-{\left(A_2\right)}_Iϵ\\ {\left(A_4\right)}_{st}+{\left(A_4\right)}_Iϵ& -{\left(A_3\right)}_{st}-{\left(A_3\right)}_Iϵ& {\left(A_2\right)}_{st}+{\left(A_2\right)}_Iϵ& {\left(A_1\right)}_{st}+{\left(A_1\right)}_Iϵ\end{array}\right].\end{array}$$

The first column block of $\psi \left(A\right)$ is

$$\begin{array}{c}\hfill \psi {\left(A\right)}^c=\left[\begin{array}{c}{\left(A_1\right)}_{st}+{\left(A_1\right)}_Iϵ\\ {\left(A_2\right)}_{st}+{\left(A_2\right)}_Iϵ\\ {\left(A_3\right)}_{st}+{\left(A_3\right)}_Iϵ\\ {\left(A_4\right)}_{st}+{\left(A_4\right)}_Iϵ\end{array}\right],\end{array}$$

the first row block of $\psi \left(A\right)$ is

$$\begin{array}{c}\hfill \psi {\left(A\right)}^r=\left[\begin{array}{cccc}{\left(A_1\right)}_{st}+{\left(A_1\right)}_Iϵ& -{\left(A_2\right)}_{st}-{\left(A_2\right)}_Iϵ& -{\left(A_3\right)}_{st}-{\left(A_3\right)}_Iϵ& -{\left(A_4\right)}_{st}-{\left(A_4\right)}_Iϵ\end{array}\right].\end{array}$$

It can be proved that the dual matrix representation $\psi \left(A\right)$ and the first column $\psi {\left(A\right)}^c$/row block $\psi {\left(A\right)}^r$ have the following properties.

Theorem 1.

Let A, $B\in {\mathbb{DQ}}^{m\times n}$, $C\in {\mathbb{DQ}}^{n\times p}$, $\lambda \in \mathbb{D}$, then

• $\left(1\right)\psi (A+B)=\psi \left(A\right)+\psi \left(B\right);\psi \left(\lambda A\right)=\lambda \psi \left(A\right);\psi \left(AC\right)=\psi \left(A\right)\psi \left(C\right).$
• $\left(2\right)\psi {(A+B)}^c=\psi {\left(A\right)}^c+\psi {\left(B\right)}^c;\psi {\left(\lambda A\right)}^c=\lambda \psi {\left(A\right)}^c;\psi {\left(AC\right)}^c=\psi \left(A\right)\psi {\left(C\right)}^c.$
• $\left(3\right)\psi {(A+B)}^r=\psi {\left(A\right)}^r+\psi {\left(B\right)}^r;\psi {\left(\lambda A\right)}^r=\lambda \psi {\left(A\right)}^r;\psi {\left(AC\right)}^r=\psi {\left(A\right)}^r\psi \left(C\right).$

Similar to the 2-norm of the dual quaternion vector, we give the definition of the 2-norm of the dual vector.

Definition 2.

For any $x\in {\mathbb{D}}^n$, the 2-norm of x is

$${\parallel x\parallel }_{2\left(D\right)}=\left\{\begin{array}{c}\sqrt{{\displaystyle \sum _{i=1}^n}{\left|x_i\right|}^2},if{x}_{st}\ne {\mathbb{O}}^n,\hfill \\ \sqrt{{\displaystyle \sum _{i=1}^n}{\left|{\left(x_i\right)}_I\right|}^2}ϵ,if{x}_{st}={\mathbb{O}}^n.\hfill \end{array}\right.$$

Next, we propose the relationship between the 2-norm of the dual quaternion vector and the 2-norm of the first column block of its dual representation, which will help us to solve Problem 1.

Theorem 2.

Suppose $x=x_{st}+x_Iϵ\in {\mathbb{DQ}}^n$, where $x_{st}={[x_{11},\dots ,x_{n1}]}^T$, $x_I={[x_{12},\dots ,x_{n2}]}^T$ and $x_{ml}={\left(x_1\right)}_{ml}+{\left(x_2\right)}_{ml}i+{\left(x_3\right)}_{ml}j+{\left(x_4\right)}_{ml}k$, $m=1,\dots ,n,l=1,2.$ Then

$${\parallel x\parallel }_{2\left(DQ\right)}={\parallel \psi {\left(x\right)}^c\parallel }_{2\left(D\right)}.$$

Proof. 

(1). If $x_{st}={\mathbb{O}}^n$, then

$$x=\left[\begin{array}{c}{x}_{12}ϵ\\ ⋮\\ x_{n2}ϵ\end{array}\right],\psi {\left(x\right)}^c=\left[\begin{array}{c}{\left(x_1\right)}_{12}ϵ\\ ⋮\\ {\left(x_1\right)}_{n2}ϵ\\ ⋮\\ {\left(x_4\right)}_{12}ϵ\\ ⋮\\ {\left(x_4\right)}_{n2}ϵ\end{array}\right].$$

Thus, we obtain

$$\begin{array}{cc}\hfill {\parallel x\parallel }_{2\left(DQ\right)}=& \sqrt{\sum _{i=1}^n{\left|x_{i2}\right|}^2ϵ}\hfill \\ \hfill =& \sqrt{\sum _{i=1}^n{|{\left(x_1\right)}_{i2}+{\left(x_2\right)}_{i2}i+{\left(x_3\right)}_{i2}j+{\left(x_4\right)}_{i2}k|}^2ϵ}\hfill \\ \hfill =& \sqrt{\sum _{i=1}^n({\left(x_1\right)}_{i2}^2+{\left(x_2\right)}_{i2}^2+{\left(x_3\right)}_{i2}^2+{\left(x_4\right)}_{i2}^2)ϵ}\hfill \\ \hfill =& {\parallel \psi \left(x\right)}^c{\parallel }_{2\left(D\right)}.\hfill \end{array}$$

(2). If $x_{st}\ne {\mathbb{O}}^n$, we suppose $x_{11},\dots ,x_{(s-1)1}\ne 0$, $x_{s1},\dots ,x_{n1}=0$, $1<s\le n$

$$\begin{array}{cc}\hfill {\parallel x\parallel }_{2\left(DQ\right)}=& \sqrt{\sum _{i=1}^{s-1}|x_{i1}+x_{i2}{ϵ|}^2+\sum _{i=s}^n{|x_{i2}|}^2}\hfill \\ \hfill =& \sqrt{\sum _{i=1}^{s-1}|\sum _{p=1}^4({\left(x_p\right)}_{i1}+{\left(x_p\right)}_{i2}ϵ){|}^2+\sum _{i=s}^n{|x_{i2}|}^2}\hfill \\ \hfill =& {\parallel \psi \left(x\right)}^c{\parallel }_{2\left(D\right)},\hfill \end{array}$$

To sum up, we can obtain ${\parallel x\parallel }_{2\left(DQ\right)}={\parallel \psi {\left(x\right)}^c\parallel }_{2\left(D\right)}$.   □

2.2. The STP of Dual Quaternion Matrices

Based on the concept of STP of real matrices [25] and STP of quaternion matrices [26], we will define the STP of dual quaternion matrices in this section and propose some properties.

Definition 3.

Suppose $A\in {\mathbb{DQ}}^{m\times n}$, $B\in {\mathbb{DQ}}^{p\times q}$, and $t=lcm(n,p)$ is the least common multiple of n and p. Then the left STP of A and B is defined by

$$A⋉B=\left(A\otimes I_{t/n}\right)\left(B\otimes I_{t/p}\right).$$

The right STP of A and B is defined by

$$A⋊B=\left(I_{t/n}\otimes A\right)\left(I_{t/p}\otimes B\right).$$

If $n=p$, the left and right STP of dual quaternion matrices reduces to the conventional matrix product. We use the symbol ⋈ to express the left and right STP. For example, $A\bowtie B$ means $A⋉B$ and $A⋊B$.

According to the definition, we can easily obtain the following basic properties of the STP of dual quaternion matrices.

Theorem 3.

Let A, B, C, D be dual quaternion matrices and a, b be dual numbers, then

• $\left(a\right)A\bowtie (aB+bC)=aA\bowtie B+bA\bowtie C$, $(aA+bB)\bowtie C=aA\bowtie C+bB\bowtie C$,
• $\left(b\right)A\bowtie \left(B\bowtie C\right)=\left(A\bowtie B\right)\bowtie C$,
• $\left(c\right){\left(A\bowtie B\right)}^H=B^H\bowtie A^H$.

Definition 4.

For $A\in {\mathbb{DQ}}^{m\times n}$, $Co{l}_j\left(A\right)(1\le j\le n)$ and $Ro{w}_i\left(A\right)(1\le i\le m)$ represent the j-th column and the i-th row of A, respectively. Denote

$$\begin{array}{cc}\hfill & V_c\left(A\right)={[{\left(Co{l}_1\left(A\right)\right)}^T,{\left(Co{l}_2\left(A\right)\right)}^T,\dots ,{\left(Co{l}_n\left(A\right)\right)}^T]}^T,\hfill \\ \hfill & V_r\left(A\right)={[Ro{w}_1\left(A\right),Ro{w}_2\left(A\right),\dots ,Ro{w}_m\left(A\right)]}^T.\hfill \end{array}$$

$V_c\left(A\right)$ and $V_r\left(A\right)$ are called the column vector representation and the row vector representation of dual quaternion matrix A, respectively.

The swap matrix, which has an important role in STP theory, is defined as follows.

Definition 5

([25]). The swap matrix is defined as

$$\begin{array}{cc}\hfill W_{[m,n]}& =[I_n\otimes {\delta }_m^1,I_n\otimes {\delta }_m^2,\dots ,I_n\otimes {\delta }_m^m].\hfill \end{array}$$

By using the swap matrix, the following properties of the dual quaternion matrix or vector can be obtained.

Theorem 4.

Let $A\in {\mathbb{DQ}}^{m\times n}$, $x\in {\mathbb{DQ}}^m$, $y\in {\mathbb{DQ}}^n$, $p\in {\mathbb{DQ}}_m$, $q\in {\mathbb{DQ}}_n$, then

• $\left(a\right)W_{[m,n]}⋉\overline{x⋉y}=\overline{y}⋉\overline{x}$,
• $\left(b\right)\overline{p⋉q}⋉W_{[m,n]}=\overline{q}⋉\overline{p},$
• $\left(c\right)W_{[m,n]}{V}_r\left(A\right)=V_c\left(A\right)$, $W_{[n,m]}{V}_c\left(A\right)=V_r\left(A\right),$
• $\left(d\right)\overline{x^{T}A}=V_r^T\left(\overline{A}\right)⋉\overline{x}$, $\overline{Ay}=y^H⋉V_c\left(\overline{A}\right).$

Proof. 

Let $x={\delta }_m^i$, $y={\delta }_n^j$, then

$$x⋉y={\delta }_{mn}^{(i-1)n+j},$$

so

$$W_{[m,n]}⋉x⋉y=Co{l}_{(i-1)n+j}\left(W_{[m,n]}\right).$$

From Definition 5, it can be seen that the $\left(\right(i-1)n+j)$-th column of $W_{[m,n]}$ is the j-th column of its i-th block. Since the i-th block of $W_{[m,n]}$ is

$$I_n\otimes {\delta }_m^i=[{\delta }_n^1\otimes {\delta }_m^i,{\delta }_n^2\otimes {\delta }_m^i,\dots ,{\delta }_n^n\otimes {\delta }_m^i].$$

The j-th column of $I_n\otimes {\delta }_m^i$ is

$${\delta }_n^j\otimes {\delta }_m^i={\delta }_n^j⋉{\delta }_m^i.$$

Therefore,

$$\begin{array}{c}\hfill W_{[m,n]}⋉{\delta }_m^i⋉{\delta }_n^j={\delta }_n^j⋉{\delta }_m^i.\end{array}$$

(2)

Let $x={[x_1,x_2,\dots ,x_m]}^T$, $y={[y_1,y_2,\dots ,y_n]}^T$ be arbitrary dual quaternion column vectors. Using (2), we have

$$\begin{array}{cc}\hfill W_{[m,n]}⋉\overline{x⋉y}& =W_{[m,n]}\overline{\left(\sum _{i=1}^m{x}_i{\delta }_m^i\right)⋉\left(\sum _{j=1}^n{y}_j{\delta }_n^j\right)}\hfill \\ \hfill & =W_{[m,n]}\left(\sum _{i=1}^m\sum _{j=1}^n\overline{x_i{y}_j}\left({\delta }_m^i{\delta }_n^j\right)\right)\hfill \\ \hfill & =\sum _{i=1}^m\sum _{j=1}^n\overline{x_i{y}_j}\left(W_{[m,n]}({\delta }_m^i⋉{\delta }_n^j)\right)\hfill \\ \hfill & =\sum _{i=1}^m\sum _{j=1}^n\overline{x_i{y}_j}({\delta }_n^j⋉{\delta }_m^i)\hfill \\ \hfill & =\overline{y}⋉\overline{x}.\hfill \end{array}$$

Using the definitions of the swap matrix and vector operator, it is easy to verify $\left(b\right)$, $\left(c\right)$, $\left(d\right)$.   □

The next two theorems are the core of transforming the QLY least squares problem of the dual quaternion matrix equation into the QLY least squares problem of a dual linear system. For the proof of the properties of vector operators on quaternion matrices, please refer to [26].

Theorem 5.

Let $A\in {\mathbb{DQ}}^{m\times n}$, $X\in {\mathbb{DQ}}^{n\times p}$, then

$$V_r\left(AX\right)=A⋉V_r\left(X\right),$$

$$V_c\left(AX\right)=A⋊V_c\left(X\right).$$

Proof. 

For $A=A_{st}+A_Iϵ\in {\mathbb{DQ}}^{m\times n}$, $X=X_{st}+X_Iϵ\in {\mathbb{DQ}}^{n\times p}$, then

$$\begin{array}{cc}\hfill V_r\left(AX\right)& =V_r\left(A_{st}{X}_{st}\right)+V_r\left((A_{st}{X}_I+A_I{X}_{st})ϵ\right)\hfill \\ \hfill & =A_{st}⋉V_r\left(X_{st}\right)+A_{st}⋉V_r\left(X_Iϵ\right)+A_Iϵ⋉V_r\left(X_{st}\right)\hfill \\ \hfill & =(A_{st}+A_Iϵ)⋉V_r\left(X_{st}+X_Iϵ\right)\hfill \\ \hfill & =A⋉V_r\left(X\right).\hfill \end{array}$$

By using the swap matrix for $V_r\left(AX\right)=A⋉V_r\left(X\right)$, we can obtain

$$\begin{array}{cc}\hfill V_c\left(AX\right)& =W_{[m,p]}⋉V_r\left(AX\right)\hfill \\ & =W_{[m,p]}⋉A⋉V_r\left(X\right)\hfill \\ \hfill & =W_{[m,p]}⋉A⋉W_{[p,n]}⋉V_c\left(X\right)\hfill \\ \hfill & =(I_p\otimes A)V_c\left(X\right)\hfill \\ & =A⋊V_c\left(X\right).\hfill \end{array}$$

□

Theorem 6.

Let $B\in {\mathbb{DQ}}^{m\times n}$, $Y\in {\mathbb{DQ}}^{q\times m}$, then

$$V_c\left(\overline{YB}\right)=B^H⋉V_c\left(\overline{Y}\right),$$

$$V_r\left(\overline{YB}\right)=B^H⋊V_r\left(\overline{Y}\right).$$

Proof. 

For $B=B_{st}+B_Iϵ$, $Y=Y_{st}+Y_iϵ$, we have

$$\begin{array}{cc}\hfill {\overline{YB}}^T& ={\left(\overline{Y_{st}{B}_{st}}\right)}^T+{\left(\overline{Y_{st}{B}_I+Y_I{B}_{st}}\right)}^Tϵ\hfill \\ \hfill & =B_{st}^H{Y}_{st}^H+(B_{st}^H{Y}_I^H+B_I^H{Y}_{st}^H)ϵ\hfill \\ \hfill & =B^H{Y}^H.\hfill \end{array}$$

By Theorem 5, then

$$V_c\left(\overline{YB}\right)=V_r\left(B^H{Y}^H\right)=B^H⋉V_r\left(Y^H\right)=B^H⋉V_c\left(\overline{Y}\right)$$

By using the swap matrix for $V_r\left(\overline{YB}\right)=B^H⋉V_c\left(\overline{Y}\right)$, we can obtain

$$\begin{array}{cc}\hfill V_r\left(\overline{YB}\right)& =W_{[n,q]}⋉V_c\left(\overline{YB}\right)\hfill \\ \hfill & =W_{[n,q]}⋉B^H⋉V_c\left(\overline{Y}\right)\hfill \\ \hfill & =W_{[n,q]}⋉B^H⋉W_{[q,m]}⋉V_r\left(\overline{Y}\right)\hfill \\ \hfill & =(I_q\otimes B^H)V_r\left(\overline{Y}\right)\hfill \\ \hfill & =B^H⋊V_r\left(\overline{Y}\right).\hfill \end{array}$$

□

3. Algebraic Solution to Problem 1

We first introduce the H-representation of special real matrices, which will reduce the complexity of solving matrix equations. Based on this, we can generalize the H-representation to dual quaternion matrices with special structures, which we call $\mathcal{GH}$-representation.

Definition 6

([27]). Consider a p-dimensional real matrix subspace $\mathbb{X}\subset {\mathbb{R}}^{n\times n}$. Assume $e_1$, $e_2$, …, $e_p$ form the bases of $\mathbb{X}$, which means that for any $X\in \mathbb{X}$, we have $X=x_1{e}_1+x_2{e}_2+\dots +x_p{e}_p$, and define $H=[V_r\left(e_1\right),V_r\left(e_2\right),\dots ,V_r\left(e_p\right)]$. For each $X\in \mathbb{X}$, if we express

$$V_r\left(X\right)=H\tilde{X}$$

and $\tilde{X}={[x_1,x_2,\dots ,x_p]}^T$, then $H\tilde{X}$ is called an H-representation of $V_r\left(X\right)$, and H is called an H-representation matrix of $V_r\left(X\right)$.

For the real symmetric matrix and the real antisymmetric matrix, the following conclusion can be obtained by using the H-representation.

Theorem 7.

Let $X_1\in {\mathbb{R}}_S^{n\times n}$, $X_2\in {\mathbb{R}}_A^{n\times n}$. Then

$$\begin{array}{c}\hfill V_r\left(X_1\right)=H_1\widetilde{X_1},\\ \hfill V_r\left(X_2\right)=H_2\widetilde{X_2},\end{array}$$

where

$$\begin{array}{c}\hfill H_1=(V_r\left(P_{11}\right),\dots ,V_r\left(P_{1n}\right),V_r\left(P_{22}\right),\dots ,V_r\left(P_{2n}\right),\dots ,V_r\left(P_{nn}\right)),\end{array}$$

and $P_{ij}={\left(p_{lk}\right)}_{n\times n}$ with $p_{ij}=p_{ji}=1$, the other entries being zero, and

$$\begin{array}{c}\hfill \widetilde{X_1}={(x_{11},\dots ,x_{1n},x_{22},\dots ,x_{2n},\dots ,x_{nn})}^T.\end{array}$$

$$\begin{array}{c}\hfill H_2=(V_r\left(Q_{12}\right),\dots ,V_r\left(Q_{1n}\right),V_r\left(Q_{23}\right),\dots ,V_r\left(Q_{2n}\right),\dots ,V_r\left(Q_{(n-1),n}\right)),\end{array}$$

and $Q_{ij}={\left(q_{lk}\right)}_{n\times n}$ with $q_{ij}=-q_{ji}=1$, the other entries being zero, and

$$\begin{array}{c}\hfill \widetilde{X_2}={(x_{12},\dots ,x_{1n},x_{23},\dots ,x_{2n},\dots ,x_{(n-1),n})}^T.\end{array}$$

The H-representation provides a method for extracting independent elements of a special structure matrix.

Definition 7.

Consider a p-dimensional dual quaternion matrix subspace $\mathbb{Y}\subset {\mathbb{DQ}}^{n\times n}$. For each $X=X_1+X_{2}i+X_{3}j+X_{4}k\in \mathbb{Y}$, where $X_m={\left(X_m\right)}_{st}+{\left(X_m\right)}_Iϵ$, $m=1,2,3,4$, if we express

$$V_r\left(\left[\begin{array}{c}{X}_1\\ X_2\\ X_3\\ X_4\end{array}\right]\right)=H_D\widehat{X},$$

then $H_D\widehat{X}$ is called a $\mathcal{GH}$-representation of $V_r\left(\left[\begin{array}{c}{X}_1\\ X_2\\ X_3\\ X_4\end{array}\right]\right)$, where

$$H_D=\left[\begin{array}{cccc}{H}_{X_1}& & & \\ & H_{X_2}& & \\ & & H_{X_3}& \\ & & & H_{X_4}\end{array}\right], \widehat{X}=\left[\begin{array}{c}\widetilde{X_1}\\ \widetilde{X_2}\\ \widetilde{X_3}\\ \widetilde{X_4}\end{array}\right],$$

$H_{X_m}$ represents the H-representation matrix of $X_m$, $(m=1,2,3,4)$.

According to Definition 7, the $\mathcal{GH}$-represeation of the dual quaternion Hermitian matrix is given in the following Theorem.

Theorem 8.

Let $X=X_1+X_{2}i+X_{3}j+X_{4}k\in {\mathbb{DQ}}_H^{n\times n}$, where $X_m={\left(X_m\right)}_{st}+{\left(X_m\right)}_Iϵ$, $m=1,2,3,4$, then

$${\psi }^c\left(V_r\left(X\right)\right)=\left[\begin{array}{c}{V}_r\left(X_1\right)\\ V_r\left(X_2\right)\\ V_r\left(X_3\right)\\ V_r\left(X_4\right)\end{array}\right]=\left[\begin{array}{cccc}{H}_1& & & \\ & H_2& & \\ & & H_2& \\ & & & H_2\end{array}\right]\left[\begin{array}{c}\widetilde{X_1}\\ \widetilde{X_2}\\ \widetilde{X_3}\\ \widetilde{X_4}\end{array}\right]={\mathcal{H}}_{\mathcal{D}}\widehat{X},$$

where $H_1$, $H_2$ are matrices defined in Theorem 7.

Based on the QLY least squares of linear dual least squares system [23], we will provide the QLY least squares Hermitian solution of dual quaternion matrix Equation (1) in this section.

Lemma 1

([23]). For the inconsistent dual equation $Ax\approx b$, a QLY least squares solution is

$$\begin{array}{c}\hfill x_Q=x_r+\left(A_{st}^{†}{b}_I-A_{st}^{†}{A}_I{x}_r+(I_n-A_{st}^{†}{A}_{st})k\right)ϵ,\end{array}$$

(3)

where $x_r=A_{st}^{†}{b}_{st}+(I_n-A_{st}^{†}{A}_{st})h,$$k,h\in {\mathbb{R}}^n$. The minimum norm least squares solution to the dual equation $Ax\approx b$ is given by

$$\begin{array}{c}\hfill x_{Qm}=A_{st}^{†}{b}_{st}+(A_{st}^{†}{b}_I-A_{st}^{†}{A}_I{A}_{st}^{†}{b}_{st})ϵ.\end{array}$$

(4)

Based on the Lemma 1, Problem 1 is discussed below.

Theorem 9.

Let $A\in {\mathbb{DQ}}^{m\times n}$, $B\in {\mathbb{DQ}}^{n\times p}$, $C\in {\mathbb{DQ}}^{m\times q}$, $D\in {\mathbb{DQ}}^{q\times p}$, $E\in {\mathbb{DQ}}^{m\times p}$. Denote

$$U=\psi (A\otimes I_p)N_1\psi (I_n\otimes B^H)N_3,$$

$$V=\psi (C\otimes I_q)N_2\psi (I_q\otimes D^H)N_4,$$

$$N_m=diag[I_m,-I_m,-I_m,-I_m],$$

where $m=1,2,3,4$ and $I_m$ is an identity matrix of appropriate dimension.

$$M=M_{st}+M_Iϵ=\left[\begin{array}{cc}U{\mathcal{H}}_D& V{\mathcal{H}}_D\end{array}\right],$$

${\mathcal{H}}_D$ is defined in Theorem 8.

The set $J_Q$ in Problem 1 can be expressed as

$$\begin{array}{c}\hfill J_Q=\left\{(X,Y)|\left[\begin{array}{c}\widehat{X}\\ \widehat{Y}\end{array}\right]=x_r+\left(M_{st}^{†}{\psi }^c\left(V_r\left(E_I\right)\right)-M_{st}^{†}{M}_I{x}_r+(I_n-M_{st}^{†}{M}_{st})k\right)ϵ\right\},\end{array}$$

(5)

where $x_r=M_{st}^{†}{\psi }^c\left(V_r\left(E_{st}\right)\right)+(I_n-M_{st}^{†}{M}_{st})h$, $k,h\in {\mathbb{R}}^{2(n^2+q^2)-(n+q)}$.

In particular, the QLY minimal norm least squares Hermitian solution $(X^M,Y^M)$ satisfies

$$\begin{array}{c}\hfill \left[\begin{array}{c}\widehat{X^M}\\ \widehat{Y^M}\end{array}\right]=M_{st}^{†}{\psi }^c\left(V_r\left(E_{st}\right)\right)+\left(M_{st}^{†}{\psi }^c\left(V_r\left(E_I\right)\right)-M_{st}^{†}{M}_I{M}_{st}^{†}{\psi }^c\left(V_r\left(E_{st}\right)\right)\right)ϵ.\end{array}$$

Proof. 

According to the properties of the vector operator, the STP of dual quaternion matrices, the dual representation of the dual quaternion matrix and the relationship of ${\parallel ·\parallel }_F$, ${\parallel ·\parallel }_{2\left(D\right)}$, ${\parallel ·\parallel }_{2\left(DQ\right)}$, we have

$$\begin{array}{cc}\hfill & {\parallel AXB+CYD-E\parallel }_F\hfill \\ \hfill =& \parallel V_r\left(AXB\right)+V_r\left(CYD\right)-V_r{\left(E\right)\parallel }_{2\left(DQ\right)}\hfill \\ \hfill =& \parallel A⋉\overline{V_r\left(\overline{XB}\right)}+C⋉\overline{V_r\left(\overline{YD}\right)}-V_r{\left(E\right)\parallel }_{2\left(DQ\right)}\hfill \\ \hfill =& \parallel A⋉\overline{(I_n\otimes B^H)V_r\left(\overline{X}\right)}+C⋉\overline{(I_q\otimes D^H)V_r\left(\overline{Y}\right)}-V_r{\left(E\right)\parallel }_{2\left(DQ\right)}\hfill \\ \hfill =& \parallel (A\otimes I_p)\left(\overline{(I_n\otimes B^H)V_r\left(\overline{X}\right)}\right)+(C\otimes I_p)\left(\overline{(I_q\otimes D^H)V_r\left(\overline{Y}\right)}\right)-V_r\left(E\right){\parallel }_{2\left(DQ\right)}\hfill \\ \hfill =& \parallel {\psi }^c\left((A\otimes I_p)\left(\overline{(I_n\otimes B^H)V_r\left(\overline{X}\right)}\right)\right)+{\psi }^c\left((C\otimes I_p)\left(\overline{(I_q\otimes D^H)V_r\left(\overline{Y}\right)}\right)\right)-{\psi }^c\left(V_r\left(E\right)\right){\parallel }_{2\left(D\right)}\hfill \\ \hfill =& \parallel \psi (A\otimes I_p)N_1{\psi }^c\left((I_n\otimes B^H)V_r\left(\overline{X}\right)\right)+\psi (C\otimes I_p)N_2{\psi }^c\left((I_q\otimes D^H)V_r\left(\overline{Y}\right)\right)-{\psi }^c\left(V_r\left(E\right)\right){\parallel }_{2\left(D\right)}\hfill \\ \hfill =& \parallel \psi (A\otimes I_p)N_1\psi (I_n\otimes B^H)N_3{\psi }^c\left(V_r\left(X\right)\right)+\psi (C\otimes I_p)N_2\psi (I_q\otimes D^H)N_4{\psi }^c\left(V_r\left(Y\right)\right)-{\psi }^c\left(V_r\left(E\right)\right){\parallel }_{2\left(D\right)}\hfill \\ \hfill =& \parallel U{\psi }^c\left(V_r\left(X\right)\right)+V{\psi }^c\left(V_r\left(Y\right)\right)-{\psi }^c\left(V_r\left(E\right)\right){\parallel }_{2\left(D\right)}.\hfill \end{array}$$

By using Theorem 8, we obtain

$${\psi }^c\left(V_r\left(X\right)\right)={\mathcal{H}}_D\widehat{X},$$

$${\psi }^c\left(V_r\left(Y\right)\right)={\mathcal{H}}_D\widehat{Y}.$$

Therefore, we obtain

$$\begin{array}{cc}\hfill & {\parallel AXB+CYD-E\parallel }_F\hfill \\ \hfill =& \parallel U{\mathcal{H}}_D\widehat{X}+V{\mathcal{H}}_D\widehat{Y}-{\psi }^c\left(V_r\left(E\right)\right){\parallel }_{2\left(D\right)}\hfill \\ \hfill =& {∥\left[\begin{array}{cc}U{\mathcal{H}}_D& V{\mathcal{H}}_D\end{array}\right]\left[\begin{array}{c}\widehat{X}\\ \widehat{Y}\end{array}\right]-{\psi }^c\left(V_r\left(E\right)\right)∥}_{2\left(D\right)}\hfill \\ \hfill =& {∥M\left[\begin{array}{c}\widehat{X}\\ \widehat{Y}\end{array}\right]-{\psi }^c\left(V_r\left(E\right)\right)∥}_{2\left(D\right)}.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill & {∥AXB+CYD-E∥}_F=mi{n}_Q\iff {∥M\left[\begin{array}{c}\widehat{X}\\ \widehat{Y}\end{array}\right]-{\psi }^c\left(V_r\left(E\right)\right)∥}_{2\left(D\right)}=mi{n}_Q.\hfill \end{array}$$

Thus, by using Lemma 1, its QLY least squares Hermitian solution can be represented as

$$\begin{array}{c}\hfill \left[\begin{array}{c}\widehat{X}\\ \widehat{Y}\end{array}\right]=x_r+\left(M_{st}^{†}{\psi }^c\left(V_r\left(E_I\right)\right)-M_{st}^{†}{M}_I{x}_r+(I_n-M_{st}^{†}{M}_{st})k\right)ϵ,\end{array}$$

where $x_r=M_{st}^{†}{\psi }^c\left(V_r\left(E_{st}\right)\right)+(I_n-M_{st}^{†}{M}_{st})h$, $k,h\in {\mathbb{R}}^{2(n^2+q^2)-(n+q)}$. Thus, we obtain the Formula (5). And the QLY minimal norm least squares Hermitian solution $(X^M,Y^M)$ satisfies

$$\begin{array}{c}\hfill \left[\begin{array}{c}\widehat{X^M}\\ \widehat{Y^M}\end{array}\right]=M_{st}^{†}{\psi }^c\left(V_r\left(E_{st}\right)\right)+\left(M_{st}^{†}{\psi }^c\left(V_r\left(E_I\right)\right)-M_{st}^{†}{M}_I{M}_{st}^{†}{\psi }^c\left(V_r\left(E_{st}\right)\right)\right)ϵ.\end{array}$$

□

4. Numerical Algorithm and Examples

Based on the discussion in Section 3, we propose the numerical algorithm to solve Problem 1 and then give two examples to verify the feasibility of the proposed algorithm.

Algorithm 1 is based on Theorem 9. We next give numerical examples to show the effectiveness of the algorithm.    

Algorithm 1: Calculate the QLY minimal norm least squares Hermitian solution.

Input: $A\in {\mathbb{DQ}}^{m\times n},B\in {\mathbb{DQ}}^{n\times p},C\in {\mathbb{DQ}}^{m\times q},D\in {\mathbb{DQ}}^{q\times p},E\in {\mathbb{DQ}}^{m\times p}$;

Output: $\left[\begin{array}{c}\widehat{X}\\ \widehat{Y}\end{array}\right]$

1. Fix the form of $\psi$ satisfying Definition 1;

2. Calculate ${\mathcal{H}}_D$ of $\mathcal{GH}$-representation of dual quaternion Hermitian matrix;

3. Calculate U, V;

4. Calculate $M=\left[\begin{array}{cc}U{\mathcal{H}}_D& V{\mathcal{H}}_D\end{array}\right]$;

5. Calculate the QLY least squares Hermitian solution according to (5).

Example 1.

Given $A\in {\mathbb{DQ}}^{m\times n}$, $B\in {\mathbb{DQ}}^{n\times p}$, $C\in {\mathbb{DQ}}^{m\times q}$, $D\in {\mathbb{DQ}}^{q\times p}$, $E\in {\mathbb{DQ}}^{m\times p}$, where

$$\begin{array}{cc}\hfill A& =\left[\begin{array}{ccc}24+31i+44j+21k& 33+28i+45j+7k& 22+42i+24j+14k\\ 28+13i+7j+26k& 45+42i+11j+44k& 15+11i+42j+44k\\ 3+7i+43j+7k& 6+42i+4j+31k& 50+28i+24j+3k\end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{ccc}23+19i+36j+12k& 21+12i+4j+4k& 39+45i+42j+21k\\ 37+44i+38j+27k& 27+30i+17j+5k& 4+47i+26j+7k\\ 17+21i+45j+15k& 45+24i+1j+4k& 14+41i+19j+23k\end{array}\right]ϵ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill B& =\left[\begin{array}{ccc}45+38i+24j+26k& 29+35i+36j+47k& 3+44i+33j+13k\\ 18+23i+8j+25k& 44+29i+16j+20k& 8+34i+27j+35k\\ 7+34i+4j+25k& 18+32i+34j+6k& 4+44i+36j+42k\end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{ccc}49+16i+50j+47k& 30+20i+28j+17k& 25+6i+34j+19k\\ 11+5i+44j+29k& 21+39i+50j+37k& 33+46i+4j+47k\\ 39+42i+2j+22k& 26+9i+26j+1k& 38+44i+50j+28k\end{array}\right]ϵ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill C& =\left[\begin{array}{ccc}24+11i+30j+7k& 48+37i+35j+21k& 50+26i+21j+34k\\ 25+10i+8j+9k& 50+27i+31j+25k& 23+28i+32j+1k\\ 16+42i+3j+1k& 26+42i+11j+8k& 22+11i+28j+6k\end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{ccc}48+1i+43j+27k& 25+35i+18j+8k& 42+13i+13j+24k\\ 49+19i+15j+41k& 44+19i+50j+22k& 41+29i+41j+30k\\ 2+46i+35j+37k& 13+32i+5j+18k& 32+50i+5j+8k\end{array}\right]ϵ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill D& =\left[\begin{array}{ccc}46+29i+32j+3k& 29+46i+20j+20k& 35+47i+31j+45k\\ 33+13i+46j+27k& 47+45i+38j+41k& 24+9i+5j+36k\\ 9+16i+34j+6k& 40+40i+41j+13k& 14+26i+47j+19k\end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{ccc}13+14i+16j+48k& 3+24i+40j+26k& 38+10i+3j+4k\\ 13+44i+37j+5k& 35+5i+11j+31k& 49+27i+41j+43k\\ 39+41i+26j+46k& 32+29i+34j+16k& 20+18i+34j+8k\end{array}\right]ϵ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill E& =\left[\begin{array}{ccc}19+38i+37j+45k& 32+22i+32j+24k& 50+41i+4j+29k\\ 26+19i+7j+25k& 50+16i+50j+4k& 12+50i+25j+45k\\ 20+36i+2j+15k& 35+21i+3j+14k& 4+5i+8j+30k\end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{ccc}24+14i+35j+47k& 19+44i+19j+26k& 33+41i+4j+13k\\ 47+24i+23j+13k& 32+5i+3j+38k& 15+29i+37j+50k\\ 11+10i+2j+18k& 22+27i+48j+38k& 2+18i+38j+6k\end{array}\right]ϵ.\hfill \end{array}$$

By using Algorithm 1, it is easy to obtain the QLY minimal norm least squares Hermitian solution

$$X=[{\alpha }_1,{\alpha }_2,{\alpha }_3]+[{\alpha }_4,{\alpha }_5,{\alpha }_6]ϵ,Y=[{\beta }_1,{\beta }_2,{\beta }_3]+[{\beta }_4,{\beta }_5,{\beta }_6]ϵ,$$

where

$${\alpha }_1=\left[\begin{array}{c}-0.0070+0i+0j+0k\\ -0.0053+0.0037i-0.0016j+0.0040k\\ 0.0030-0.0073i+0.0012j-0.0055k\end{array}\right],{\alpha }_2=\left[\begin{array}{c}-0.0053-0.0037i+0.0016j-0.0040k\\ 0.0095+0i+0j+0k\\ -0.0021+0.0028i-0.0058j+0.0007k\end{array}\right],$$

$${\alpha }_3=\left[\begin{array}{c}0.0030+0.0073i-0.0012j+0.0055k\\ -0.0021-0.0028i+0.0058j-0.0007k\\ -0.0045+0i+0j+0k\end{array}\right],{\alpha }_4=\left[\begin{array}{c}0.0097+0i+0j+0k\\ -0.0044-0.0032i+0.0052j+0.0055k\\ 0.0201+0.0039i+0.0074j-0.0166k\end{array}\right],$$

$${\alpha }_5=\left[\begin{array}{c}-0.0044+0.0032i-0.0052j-0.0055k\\ 0.0294+0i+0j+0k\\ -0.0118-0.0101i-0.0201j+0.0064k\end{array}\right],{\alpha }_6=\left[\begin{array}{c}0.0201-0.0039i-0.0074j+0.0166k\\ -0.0118+0.0101i+0.0201j-0.0064k\\ -0.0221+0i+0j+0k\end{array}\right],$$

and

$${\beta }_1=\left[\begin{array}{c}0.0215+0i+0j+0k\\ -0.0078+0.0002i+0.0084j+0.0009k\\ 0.0067-0.0028i-0.0153j+0.0042k\end{array}\right],{\beta }_2=\left[\begin{array}{c}-0.0078-0.0002i-0.0084j-0.0009k\\ 0.0277+0i+0j+0k\\ -0.0190+0.0122i-0.0024j+-0.0018k\end{array}\right],$$

$${\beta }_3=\left[\begin{array}{c}0.0067+0.0028i+0.0153j-0.0042k\\ -0.0190-0.0122i+0.0024j+0.0018k\\ 0.0212+0i+0j+0k\end{array}\right],{\beta }_4=\left[\begin{array}{c}0.0097+0i+0j+0k\\ -0.0065+0.0317i-0.0220j-0.0266k\\ 0.0361-0.0460i+0.0058j+0.0666k\end{array}\right],$$

$${\beta }_5=\left[\begin{array}{c}-0.0065-0.0317i+0.0220j+0.0266k\\ -0.0629+0i+0j+0k\\ 0.0298+0.0412i-0.0002j-0.0570k\end{array}\right],{\beta }_6=\left[\begin{array}{c}0.0361+0.0460i-0.0058j-0.0666k\\ 0.0298-0.0412i+0.0002j+0.0570k\\ -0.0576+0i+0j+0k\end{array}\right].$$

In Example 1, we show a numerical example to solve the Hermitian solution for the QLY minimal norm least squares problem of dual quaternion matrix Equation (1), which has a special structure. For the solution without a special structure, we only need to omit the ${\mathcal{H}}_D$ in Algorithm 1 according to the $\mathcal{GH}$-representation, and then we can obtain the solution without a special structure. A numerical example is given below.

Example 2.

Given $A\in {\mathbb{DQ}}^{m\times n}$, $B\in {\mathbb{DQ}}^{n\times p}$, $C\in {\mathbb{DQ}}^{m\times q}$, $D\in {\mathbb{DQ}}^{q\times p}$, $E\in {\mathbb{DQ}}^{m\times p}$, where

$$\begin{array}{c}\hfill A=\left[\begin{array}{cc}4+4i+4j+2k& 2+1i+5j+2k\\ 4+5i+4j+4k& 3+1i+1j+5k\end{array}\right]+\left[\begin{array}{cc}2+5i+1j+2k& 4+3i+5j+3k\\ 1+4i+2j+2k& 2+1i+4j+5k\end{array}\right]ϵ,\end{array}$$

$$\begin{array}{c}\hfill B=\left[\begin{array}{cc}5+1i+3j+3k& 4+1i+1j+2k\\ 3+3i+1j+2k& 4+3i+5j+1k\end{array}\right]+\left[\begin{array}{cc}1+3i+4j+5k& 2+1i+4j+4k\\ 2+3i+5j+2k& 4+5i+5j+4k\end{array}\right]ϵ,\end{array}$$

$$\begin{array}{c}\hfill C=\left[\begin{array}{cc}3+1i+2j+4k& 1+3i+1j+2k\\ 5+1i+2j+3k& 1+3i+5j+5k\end{array}\right]+\left[\begin{array}{cc}5+5i+3j+4k& 3+2i+5j+5k\\ 5+1i+3j+3k& 5+5i+5j+3k\end{array}\right]ϵ,\end{array}$$

$$\begin{array}{c}\hfill D=\left[\begin{array}{cc}1+1i+5j+5k& 2+1i+5j+5k\\ 2+1i+4j+1k& 3+5i+5j+3k\end{array}\right]+\left[\begin{array}{cc}1+1i+1j+3k& 5+5i+1j+5k\\ 2+4i+2j+2k& 5+2i+3j+3k\end{array}\right]ϵ,\end{array}$$

$$\begin{array}{c}\hfill E=\left[\begin{array}{cc}2+1i+3j+1k& 1+4i+2j+3k\\ 2+1i+4j+1k& 2+4i+2j+5k\end{array}\right]+\left[\begin{array}{cc}4+1i+3j+3k& 5+4i+5j+5k\\ 4+5i+1j+4k& 5+2i+3j+5k\end{array}\right]ϵ.\end{array}$$

By using Algorithm 1, we can obtain the QLY minimal norm least squares solution

$$\begin{array}{cc}\hfill X& =\left[\begin{array}{cc}0.0115-0.0128i+0.0030j-0.0161k& 0.0203-0.0170i+0.0048j+0.0100k\\ -0.0196-0.0008i-0.0130j-0.0215k& 0.0277+0.0050i-0.0431j-0.0370k\end{array}\right]\hfill \\ & +\left[\begin{array}{cc}-0.0288-0.0429i+0.0014j+0.0192k& 0.0197-0.0179i+0.0306j+0.0251k\\ -0.1377-0.0483i+0.0294j-0.0851k& 0.0576+0.0988i-0.0356j+0.0310k\end{array}\right]ϵ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill Y& =\left[\begin{array}{cc}0.0196-0.0207i+0.0171j-0.0097k& 0.0274+0.0168i-0.0119j-0.0113k\\ 0.0284+0.0025i-0.0114j+0.0066k& -0.0417+0.0035i-0.0185j+0.0135k\end{array}\right]\hfill \\ & +\left[\begin{array}{cc}0.0698-0.0556i-0.0057j+0.0333k& 0.0351+0.0299i-0.0077j-0.0406k\\ 0.0558+0.0294i-0.0197j-0.0018k& -0.1009+0.0322i+0.1070j+0.0804k\end{array}\right]ϵ.\hfill \end{array}$$

5. Discussion

In [22], Qi, Ling and Yan proposed a total order for dual numbers. Based on this, Wang et al. [23] expanded it to dual vectors and introduced a QLY total order for dual vectors. Then, they considered the QLY least squares problem of a dual matrix equation and presented a QLY least squares solution to the inconsistent dual equation $Ax=b$. Inspired by Wang’s thought, we study the QLY least squares problem of the dual quaternion matrix equation and derive the QLY least squares Hermitian solution of dual quaternion matrix Equation (1). Whether other types of total order for dual algebra can be defined is a question worth thinking about. At the same time, we can also change the $\mathcal{GH}$-representation of the dual quaternion Hermitian matrix proposed in Theorem 8 to solve other special solutions. In addition, we know that the least squares problem of matrix equation has practical applications in computer vision, image watermarking problems, color image restoration and so on. Therefore, does the QLY least squares problem of the dual quaternion matrix equation have practical application? We will continue to explore this in future work.

6. Conclusions

In this paper, we study the QLY least squares problem of dual quaternion matrix Equation (1) and obtain the QLY least squares Hermitian solution. Firstly, we define the semi-tensor product of dual quaternion matrices and obtain the relevant properties of STP of dual quaternion matrices. Then, by using the properties of the vector operator, the dual representation of dual quaternion matrices and the $\mathcal{GH}$-representation of special dual quaternion matrices, we transform the QLY least squares of linear dual quaternion least squares problem into the QLY least squares of linear dual least squares problem. Finally, by solving the transformed dual matrix equation, we obtain the QLY (minimal norm) least squares Hermitian solution of Equation (1). The numerical examples are given to verify the effectiveness of the method.