The Least-Norm Solution to a Matrix Equation over the Dual Quaterion Algebra

Abstract

In this paper, we explore the least-norm solution to the classical matrix equation $AXB=C$ over the dual quaternion algebra, where A, B, and C are given matrices, while X remains the unknown matrix. We begin by transforming the definition of the Frobenius norm for dual quaternion matrices into an equivalent form. Using this new expression, we investigate the least-norm solution to the equation $AXB=C$ under solvability conditions. Additionally, we examine the minimum real part of the norm solution in cases where a least-norm solution does not exist. Finally, we provide two numerical examples to illustrate the main findings of our study.

1. Introduction

In 1843, William Hamilton introduced quaternions in [1], defined as

$$\mathbb{H}=\{q=q_0+q_{1}i+q_{2}j+q_{3}k|i^2=j^2=k^2=-1,ijk=-1,q_0,q_1,q_2,q_3\in \mathbb{R}\},$$

where $\mathbb{R}$ denotes the set of real numbers. The conjugate of the quaternion $q=q_0+q_{1}i+q_{2}j+q_{3}k$ is defined as $q^{*}=q_0-q_{1}i-q_{2}j-q_{3}k$. The field of quaternions and quaternion matrices has widely attracted considerable interest from mathematicians worldwide, with applications spanning mathematics, signal and color image processing, quantum physics, and computer science (e.g., [2,3,4]).

In 1873, Clifford introduced dual numbers and dual quaternions in [5]. Since then, these concepts have seen extensive applications in fields such as robotics, computer graphics, and rigid body motions (e.g., [6,7,8,9,10]). For related definitions of dual numbers and dual quaternions, please refer to Section 2.

On the other hand, matrix equations are crucial components of matrix theory. Quaternion and dual quaternion matrix equations have been applied in various fields, including control theory. However, due to the non-commutative nature of quaternion multiplication, research into quaternions and dual quaternion matrix equations is more challenging than research into real and complex matrix equations.

A substantial body of research has explored matrix equations, offering various methods for their resolution (e.g., [11,12,13,14,15]). Especially, the classical matrix equation

$$AXB=C,$$

(1)

where A, B, and C are given matrices and X is the unknown matrix, has been extensively studied. Khatri and Mitra [16] discussed the Hermitian and nonnegative definite solution of the linear matrix Equation (1). Peng [17] introduced an iterative method for the least squares symmetric solution of the linear matrix Equation (1). Liang et al. [18] provided an efficient algorithm for the generalized centro-symmetric solution of the linear matrix Equation (1). Chen et al. [19] established the necessary and sufficient conditions for the solvability of the dual quaternion matrix Equation (1). Wang et al. [20] investigated the least squares Hermitian, Persymmetric, and Bisymmetric solutions of the quaternion matrix Equation (1). Si and Wang [21] considered the general solution to the dual split quaternion matrix Equation (1). We can also obtain a more comprehensive understanding of the research on the matrix Equation (1) from reference [22], which provides a thorough overview of its solutions across various algebraic structures.

Dual quaternions provide a unified representation of rigid-body rotations and translations within the special Euclidean group SE(3) framework, effectively streamlining kinematic computations while circumventing gimbal lock limitations inherent to Euler angles. Thus, the least-norm solutions of dual quaternion matrix equations may hold potential for scientific and engineering applications that require 3D rigid-body motion optimization and energy-efficient computations, such as efficient keyframe interpolation for continuous SE(3) motion synthesis in 3D animation and the optimization of robot motion trajectories. To the best of our knowledge, the study of the least-norm solution to the matrix Equation (1) over the dual quaternion algebra remains a relatively unexplored field. This paper aims to contribute to this field by offering solvability conditions for the least-norm solution to Equation (1). Furthermore, we present an explicit expression for this solution when solvability conditions are met. In addition to these contributions, we delve into the least-norm solutions for several special cases, providing their expressions as applications.

In this paper, we use the following notations: $r\left(A\right)$ represents the rank of a quaternion matrix A, $A^{*}$ denotes the conjugate transpose of A, and tr$\left(A\right)$ stands for the trace of A. I, 0 refer to the identity matrix and the zero matrix with appropriate dimensions, respectively. The Moore–Penrose (or M-P for short) inverse of a quaternion matrix A is denoted by $A^{†}$, and it satisfies

$$A{A}^{†}A=A,A^{†}A{A}^{†}=A^{†},{\left(A{A}^{†}\right)}^{*}=A{A}^{†},{\left(A^{†}A\right)}^{*}=A^{†}A.$$

Furthermore, we use $L_A=I-A^{†}A$ and $R_A=I-A{A}^{†}$ to, respectively, represent two projectors that are induced by $A^{†}$. As usual, we use Re$\left[q\right]$ to denote the real part of a quaternion q.

The remaining contents of this paper are organized as follows. In Section 2, we review definitions of dual numbers, dual quaternions, the Frobenius norm of quaternion matrices and its equivalent expressions, and some important Lemmas for subsequent sections. In Section 3, we explore the least-norm solution to the dual quaternion matrix Equation (1) under consistency conditions and analyze the minimum real part of the norm solution when no least-norm solution exists. In Section 4, we present two numerical examples to illustrate the main findings of Section 3. In Section 5, we provide a brief summary.

2. Preliminaries

In this section, the definitions related to dual numbers, dual quaternions, and dual quaternion matrices are reviewed. Subsequently, we introduce several results concerning the norm of dual quaternion matrices, the dual quaternion matrix equation, and the equalities related to the M-P inverse of quaternion matrices.

2.1. The Definition of Dual Numbers and Its Total Order Relation

Definition 1 

([23]). The set of dual numbers is denoted as follows:

$$\mathbb{D}=\{x=x_0+x_1\epsilon |{\epsilon }^2=0and{x}_0,x_1\in \mathbb{R}\},$$

where ε is the infinitesimal unit, satisfying ${\epsilon }^2=0$ and $\epsilon \ne 0$.

In this study, $x_0$ is employed to denote the real part or the standard part of x, while $x_1$ is used to represent the dual part or the infinitesimal part of x. The infinitesimal unit $\epsilon$ commutes with real numbers, complex numbers, and quaternions under multiplication. x is designated as appreciable if $x_0\ne 0$ and otherwise classified as infinitesimal.

Assume that $x=x_0+x_1\epsilon ,y=y_0+y_1\epsilon \in \mathbb{D},k\in \mathbb{R}$; then, we have:

$$\begin{array}{cc}\hfill x+y& =x_0+y_0+(x_1+y_1)\epsilon ,\hfill \\ \hfill xy& =x_0{y}_0+(x_0{y}_1+x_1{y}_0)\epsilon ,\hfill \\ \hfill kx& =k{x}_0+k{x}_1\epsilon .\hfill \end{array}$$

Then, we define a total order relation over dual numbers. Assume $x=x_0+x_1\epsilon ,y=y_0+y_1\epsilon \in \mathbb{D}$. The relation between x and y is defined as $x=y$ if $x_0=y_0$ and $x_1=y_1$, while $x<y$ if $x_0<y_0$, or $x_0=y_0$ and $x_1<y_1$. Consequently, we call x a positive dual number if $x>0$, and x a nonnegative dual number if $x\ge 0$.

Next, we discuss the inverse element and square root of dual numbers. It is straightforward to observe that $1=1+0\epsilon$ is the unit element in $\mathbb{D}$. If x is appreciable ($x_0\ne 0$), then x is invertible and

$$x^{-1}=x_0^{-1}+x_0^{-1}{x}_1{x}_0^{-1}\epsilon ,$$

(2)

satisfying $x{x}^{-1}=x^{-1}x=1$. If x is infinitesimal, then x is not invertible.

If x is positive and appreciable, then the square root of x can be expressed as

$$\sqrt{x}=\sqrt{x_0}+{\displaystyle \frac{x_1}{2\sqrt{x_0}}}\epsilon .$$

(3)

Specifically, if $x=0$, then $\sqrt{x}=0$.

2.2. The Definition of Dual Quaternions and Dual Quaternion Matrices

Definition 2 

([23]). The set of dual quaternions is defined as follows:

$$\mathbb{DQ}=\{q=q_0+q_1\epsilon |{\epsilon }^2=0and{q}_0,q_1\in \mathbb{H}\},$$

where ε is the infinitesimal unit.

We refer to $q_0$ as the real part or the standard part of q, and $q_1$ as the dual part or the infinitesimal part of q. We call q appreciable if $q_0\ne 0$; otherwise, we call q infinitesimal. The conjugate of q is defined as $q^{*}=q_0^{*}+q_1^{*}\epsilon$.

Next, we extend Theorem 3 in reference [23] to establish the following Lemma, which reveals that the mixed product sum of two quaternions is indeed a real number.

Lemma 1. 

Suppose that $p=p_0+p_{1}i+p_{2}j+p_{3}k,q=q_0+q_{1}i+q_{2}j+q_{3}k\in \mathbb{Q}$, where $p_0,p_1,p_2,p_3,q_0,q_1,q_2,q_3\in \mathbb{R}$. Then,

$$\begin{array}{cc}\hfill p{q}^{*}+q{p}^{*}& =p^{*}q+q^{*}p=2p_0{q}_0+2p_1{q}_1+2p_2{q}_2+2p_3{q}_3\hfill \\ \hfill & =2\mathrm{Re}\left[p{q}^{*}\right]=2\mathrm{Re}\left[q{p}^{*}\right]=2\mathrm{Re}\left[p^{*}q\right]=2\mathrm{Re}\left[q^{*}p\right],\hfill \end{array}$$

which is a real number.

Proof. 

Since $q{p}^{*}={\left(p{q}^{*}\right)}^{*}$ and $q^{*}p={\left(p^{*}q\right)}^{*}$, we obtain:

$$\begin{array}{c}p{q}^{*}+q{p}^{*}=2\mathrm{Re}\left[p{q}^{*}\right]=2\mathrm{Re}\left[q{p}^{*}\right]=2p_0{q}_0+2p_1{q}_1+2p_2{q}_2+2p_3{q}_3,\\ p^{*}q+q^{*}p=2\mathrm{Re}\left[p^{*}q\right]=2\mathrm{Re}\left[q^{*}p\right]=2p_0{q}_0+2p_1{q}_1+2p_2{q}_2+2p_3{q}_3.\end{array}$$

Thus,

$$\begin{array}{cc}\hfill p{q}^{*}+q{p}^{*}& =p^{*}q+q^{*}p=2p_0{q}_0+2p_1{q}_1+2p_2{q}_2+2p_3{q}_3\hfill \\ \hfill & =2\mathrm{Re}\left[p{q}^{*}\right]=2\mathrm{Re}\left[q{p}^{*}\right]=2\mathrm{Re}\left[p^{*}q\right]=2\mathrm{Re}\left[q^{*}p\right].\hfill \end{array}$$

□

According to Lemma 1, the magnitude of $q\in \mathbb{DQ}$ can be defined as follows:

$$\left|q\right|:=\left\{\begin{array}{c}|q_0|+{\displaystyle \frac{q_0{q}_1^{*}+q_1{q}_0^{*}}{2|q_0|}}\epsilon ,\mathrm{if}{q}_0\ne 0,\hfill \\ |q_1|\epsilon ,\mathrm{otherwise},\hfill \end{array}\right.$$

(4)

which is a dual number.

Next, we introduce dual quaternion matrices and discuss their norm.

Definition 3 

([23]). The set of dual quaternion matrices is represented as follows:

$${\mathbb{DQ}}^{m\times n}=\{X=X_0+X_1\epsilon |{\epsilon }^2=0and{X}_0,X_1\in {\mathbb{H}}^{m\times n}\},$$

where ε is the infinitesimal unit.

Suppose that $X=X_0+X_1\epsilon ,Y=Y_0+Y_1\epsilon \in {\mathbb{DQ}}^{m\times n}$, and $Z=Z_0+Z_1\epsilon \in {\mathbb{DQ}}^{n\times l}$; then, we have the following:

$$\begin{array}{cc}\hfill X+Y& =X_0+Y_0+(X_1+Y_1)\epsilon ,\hfill \\ \hfill XZ& =X_0{Z}_0+(X_0{Z}_1+X_1{Z}_0)\epsilon ,\hfill \\ \hfill X^{*}& =X_0^{*}+X_1^{*}\epsilon .\hfill \end{array}$$

Before defining the norm of dual quaternion matrices, we first introduce the norm of quaternion matrices.

Definition 4 

([24]). Let $X=\left(x_{ij}\right)\in {\mathbb{H}}^{m\times n}$; the Frobenius norm of X is defined as

$$\left|\right|X\left|\right|:=\sqrt{\sum _{i=1}^m\sum _{j=1}^n{\left|x_{ij}\right|}^2}=\sqrt{\sum _{i=1}^m\sum _{j=1}^n{x}_{ij}^{*}{x}_{ij}}=\sqrt{\mathrm{tr}{X}^{*}X}.$$

(5)

Next, we introduce the definition of the norm of dual quaternion matrices.

Definition 5 

([24]). Assume $X=X_0+X_1\epsilon =\left(x_{ij}\right)\in {\mathbb{DQ}}^{m\times n}$; the Frobenius norm of X is defined as

$$\left|\right|X\left|\right|:=\left\{\begin{array}{c}\sqrt{{\sum }_{i=1}^m{\sum }_{j=1}^n{\left|x_{ij}\right|}^2},\mathrm{if}{X}_0\ne 0,\hfill \\ \left|\right|X_1\left|\right|\epsilon ,\mathrm{otherwise}.\hfill \end{array}\right.$$

(6)

In order to facilitate the discussion of the least-norm solution to the matrix Equation (1) in the following text, we have equivalently revised the definitions provided above.

Proposition 1. 

Suppose $X=X_0+X_1\epsilon =\left(x_{ij}\right)\in {\mathbb{DQ}}^{m\times n}$; then, we have that

$$\begin{array}{c}\hfill \left|\right|X\left|\right|=\left\{\begin{array}{c}\left|\right|X_0\left|\right|+{\displaystyle \frac{\mathrm{tr}(X_1^{*}{X}_0+X_0^{*}{X}_1)}{2\left|\right|X_0\left|\right|}}\epsilon ,\mathrm{if}{X}_0\ne 0,\hfill \\ \left|\right|X_1\left|\right|\epsilon ,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

(7)

$$\begin{array}{c}\hfill =\left\{\begin{array}{c}\sqrt{\mathrm{tr}{X}_0^{*}{X}_0}+{\displaystyle \frac{\mathrm{Re}\left[\mathrm{tr}{X}_1^{*}{X}_0\right]}{\sqrt{\mathrm{tr}{X}_0^{*}{X}_0}}}\epsilon ,\mathrm{if}{X}_0\ne 0,\hfill \\ \sqrt{\mathrm{tr}{X}_1^{*}{X}_1}\epsilon ,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

(8)

Proof. 

If $X_0\ne 0$, in accordance with (3) and Lemma 1, we have

$$\begin{array}{cc}\hfill \left|\right|X\left|\right|& =\sqrt{\sum _{i=1}^m\sum _{j=1}^n{\left|x_{ij}\right|}^2}\hfill \\ \hfill & =\sqrt{\sum _{{\left(x_{ij}\right)}_0\ne 0}\left(\right|{\left(x_{ij}\right)}_0|+{\displaystyle \frac{{\left(x_{ij}\right)}_0{\left(x_{ij}\right)}_1^{*}+{\left(x_{ij}\right)}_1{\left(x_{ij}\right)}_0^{*}}{2|{\left(x_{ij}\right)}_0|}}{\epsilon )}^2+\sum _{{\left(x_{ij}\right)}_0=0}\left(\right|{\left(x_{ij}\right)}_1{\left|\epsilon \right)}^2}\hfill \\ \hfill & =\sqrt{\sum _{{\left(x_{ij}\right)}_0\ne 0}\left(\right|{(x_{ij})}_0{|}^2+({\left(x_{ij}\right)}_0{\left(x_{ij}\right)}_1^{*}+{\left(x_{ij}\right)}_1{\left(x_{ij}\right)}_0^{*})\epsilon )}\hfill \\ \hfill & =\sqrt{\sum _{i=1}^m\sum _{j=1}^n(|{\left(x_{ij}\right)}_0{|}^2+({\left(x_{ij}\right)}_0{\left(x_{ij}\right)}_1^{*}+{\left(x_{ij}\right)}_1{\left(x_{ij}\right)}_0^{*})\epsilon )}\hfill \\ \hfill & =\sqrt{\sum _{i=1}^m\sum _{j=1}^n{\left|{\left(x_{ij}\right)}_0\right|}^2+\left(\sum _{i=1}^m\sum _{j=1}^n({\left(x_{ij}\right)}_0{\left(x_{ij}\right)}_1^{*}+{\left(x_{ij}\right)}_1{\left(x_{ij}\right)}_0^{*})\right)\epsilon }\hfill \\ \hfill & =\sqrt{\left|\right|X_0{\left|\right|}^2+\left(\sum _{i=1}^m\sum _{j=1}^n({\left(x_{ij}\right)}_0{\left(x_{ij}\right)}_1^{*}+{\left(x_{ij}\right)}_1{\left(x_{ij}\right)}_0^{*})\right)\epsilon }\hfill \\ \hfill & =\left|\right|X_0\left|\right|+{\displaystyle \frac{{\sum }_{i=1}^m{\sum }_{j=1}^n({\left(x_{ij}\right)}_0{\left(x_{ij}\right)}_1^{*}+{\left(x_{ij}\right)}_1{\left(x_{ij}\right)}_0^{*})}{2\left|\right|X_0\left|\right|}}\epsilon \hfill \\ \hfill & =\left|\right|X_0\left|\right|+{\displaystyle \frac{\mathrm{tr}(X_1^{*}{X}_0+X_0^{*}{X}_1)}{2\left|\right|X_0\left|\right|}}\epsilon \hfill \\ \hfill & =\sqrt{\mathrm{tr}{X}_0^{*}{X}_0}+{\displaystyle \frac{\mathrm{Re}\left[\mathrm{tr}{X}_1^{*}{X}_0\right]}{\sqrt{\mathrm{tr}{X}_0^{*}{X}_0}}}\epsilon =\sqrt{\mathrm{tr}{X}_0^{*}{X}_0}+{\displaystyle \frac{\mathrm{Re}\left[\mathrm{tr}{X}_0^{*}{X}_1\right]}{\sqrt{\mathrm{tr}{X}_0^{*}{X}_0}}}\epsilon .\hfill \end{array}$$

If $X_0=0$, since $\left|\right|X_1\left|\right|=\sqrt{\mathrm{tr}{X}_1^{*}{X}_1}$, we have $\left|\right|X\left|\right|=\left|\right|X_1\left|\right|\epsilon =\sqrt{\mathrm{tr}{X}_1^{*}{X}_1}\epsilon$. □

2.3. Some Important Lemmas

In this part, we present several Lemmas that are crucial for the subsequent proofs.

Lemma 2 

([19]). Let $A=A_0+A_1\epsilon \in {\mathbb{DQ}}^{m\times n},B=B_0+B_1\epsilon \in {\mathbb{DQ}}^{k\times l},C=C_0+C_1\epsilon \in {\mathbb{DQ}}^{m\times l}$ be known. Put

$$\begin{array}{cc}\hfill & A_2=A_1{L}_{A_0},B_2=R_{B_0}{B}_1,C_2=C_1-A_0{A}_0^{†}{C}_0{B}_0^{†}{B}_1-A_1{A}_0^{†}{C}_0{B}_0^{†}{B}_0,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & A_3=R_{A_0}{A}_2,C_{21}=R_{A_0}{C}_2,B_3=B_2{L}_{B_0},C_{22}=C_2{L}_{B_0}.\hfill \end{array}$$

(10)

Then, the dual quaternion matrix Equation (1) is consistent if and only if

$$\begin{array}{cc}\hfill & R_{A_0}{C}_0=0,C_0{L}_{B_0}=0,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & R_{A_3}{C}_{21}=0,R_{A_0}{C}_2{L}_{B_0}=0,C_{22}{L}_{B_3}=0.\hfill \end{array}$$

(12)

In this case, the general solution X of the dual quaternion matrix equation matrix Equation (1) can be expressed as $X=X_0+X_1\epsilon$, where

$$\begin{array}{cc}\hfill X_0& =A_0^{†}{C}_0{B}_0^{†}+L_{A_0}{U}_1+U_2{R}_{B_0},\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill X_1& =A_0^{†}(C_2-A_2{U}_1{B}_0-A_0{U}_2{B}_2)B_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0},\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill U_1& =A_3^{†}{C}_{21}{B}_0^{†}+L_{A_3}{Q}_1+Q_2{R}_{B_0},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill U_2& =A_0^{†}{C}_{22}{B}_3^{†}+L_{A_0}{Q}_3+Q_4{R}_{B_3},\hfill \end{array}$$

(16)

where $Q_i(i=\overline{1,4}),W_i(i=1,2)$ are arbitrary matrices over $\mathbb{H}$ with appropriate dimensions.

Lemma 3 

([25]). Let A, B, and C be arbitrary matrices over $\mathbb{H}$ with appropriate dimensions. Then, the following equalities hold:

(1) 

$A^{†}={\left(A^{*}A\right)}^{†}{A}^{*}=A^{*}{\left(A{A}^{*}\right)}^{†}$.

(2) 

$L_A={\left(L_A\right)}^2={\left(L_A\right)}^{*},R_A={\left(R_A\right)}^2={\left(R_A\right)}^{*}$.

(3) 

$L_A{\left(B{L}_A\right)}^{†}={\left(B{L}_A\right)}^{†},{\left(R_{A}C\right)}^{†}{R}_A={\left(R_{A}C\right)}^{†}$.

Lemma 4 

([25]). Assume A, B are arbitrary matrices over $\mathbb{H}$ with appropriate dimensions. Then, they satisfy the following equalities:

(1) 

$\left|\right|A+{B\left|\right|}^2={\left|\right|A\left|\right|}^2+{\left|\right|B\left|\right|}^2+2Re\left[tr{B}^{*}A\right]$.

(2) 

$Re\left[trAB\right]=Re\left[trBA\right]$.

Proposition 2. 

Given $A=\left(a_{ij}\right)\ne 0\in {\mathbb{H}}^{m\times n}$, for an arbitrary matrix $X\in {\mathbb{H}}^{m\times n}$, $\mathrm{Re}\left[\mathrm{tr}{X}^{*}A\right]$ can take any real value.

Proof. 

Since $A\ne 0$, $a_{ij}=q_0+q_{1}i+q_{2}j+q_{3}k\ne 0,q_t\in \mathbb{R},(t=0,1,2,3)$ exists. Then, we discuss two cases.

(1)

If $q_0\ne 0$, let $X=m{E}_{ij}$, where $m\in \mathbb{R}$ and $E_{ij}$ represents the matrix whose $(i,j)$ element is 1 and other elements are all 0, then we have that

$$\mathrm{Re}\left[\mathrm{tr}{X}^{*}A\right]=\mathrm{Re}\left[m{a}_{ij}\right]=m{q}_0.$$

Thus, we can change $\mathrm{Re}\left[\mathrm{tr}{X}^{*}A\right]$ to any real value through alter the value of m.

(2)

If $q_1\ne 0$ or $q_2\ne 0$ or $q_3\ne 0$, with out loss generality, suppose $q_1\ne 0$. Let $X=\left(mi\right)E_{ij}$, where $m\in \mathbb{R}$; then, we obtain that

$$\mathrm{Re}\left[\mathrm{tr}{X}^{*}A\right]=\mathrm{Re}\left[(-mi)\right)a_{ij}]=m{q}_1$$

implying that we can change $\mathrm{Re}\left[\mathrm{tr}{X}^{*}A\right]$ to any real value through alter the value of m. □

3. The Least-Norm Solution to the Matrix Equation (1) over the Dual Quaternion Algebra

In this section, starting from the general solution to the dual quaternion matrix Equation (1), we explore the least-norm solution to this matrix equation over the dual quaternion algebra.

Theorem 1. 

Assume that $A=A_0+A_1\epsilon \in {\mathbb{DQ}}^{m\times n},B=B_0+B_1\epsilon \in {\mathbb{DQ}}^{k\times l}$, and $C=C_0+C_1\epsilon \in {\mathbb{DQ}}^{m\times l}$ are given, while $X=X_0+X_1\epsilon \in {\mathbb{DQ}}^{n\times k}$ denotes the matrix to be determined. We further suppose that the dual quaternion matrix Equation (1) is consistent; let

$$\begin{array}{cc}\hfill & A_2=A_1{L}_{A_0},B_2=R_{B_0}{B}_1,C_2=C_1-A_0{A}_0^{†}{C}_0{B}_0^{†}{B}_1-A_1{A}_0^{†}{C}_0{B}_0^{†}{B}_0,\hfill \\ \hfill & A_3=R_{A_0}{A}_2,C_{21}=R_{A_0}{C}_2,B_3=B_2{L}_{B_0},C_{22}=C_2{L}_{B_0}.\hfill \end{array}$$

We consider the least-norm solution to matrix Equation (1) in three cases:

(1) 

If $A_0^{†}{C}_0{B}_0^{†}\ne 0$ and $A_3^{†}{C}_2{B}_0^{†}=A_0^{†}{C}_2{B}_3^{†}=0$, then the least-norm solution $X=X_0+X_1\epsilon$ to matrix Equation (1) exists and can be expressed as

$$\begin{array}{cc}\hfill X_0& =A_0^{†}{C}_0{B}_0^{†},\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill X_1& =A_0^{†}{C}_2{B}_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0},\hfill \end{array}$$

(18)

where $W_i(i=1,2)$ are arbitrary matrices over $\mathbb{H}$ with appropriate dimensions.

In this case, the least norm of the general solution can be represented as

$$\begin{array}{c}\hfill {\left|\right|X\left|\right|}_{min}=\left|\right|A_0^{†}{C}_0{B}_0^{†}\left|\right|+{\displaystyle \frac{\mathrm{Re}\left[\mathrm{tr}{\left(A_0^{†}{C}_2{B}_0^{†}\right)}^{*}\left(A_0^{†}{C}_0{B}_0^{†}\right)\right]}{\left|\right|A_0^{†}{C}_0{B}_0^{†}\left|\right|}}\epsilon .\end{array}$$

(2) 

Provided that $A_0^{†}{C}_0{B}_0^{†}=A_3^{†}{C}_2{B}_0^{†}=A_0^{†}{C}_2{B}_3^{†}=0$, matrix Equation (1) has a least-norm solution $X=X_0+X_1\epsilon$, which is given by

$$\begin{array}{cc}\hfill X_0& =0,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill X_1& =A_0^{†}{C}_1{B}_0^{†}.\hfill \end{array}$$

(20)

In such circumstances, the least norm of the general solution can be expressed as

$$\begin{array}{c}\hfill {\left|\right|X\left|\right|}_{min}=\left|\right|A_0^{†}{C}_1{B}_0^{†}\left|\right|\epsilon .\end{array}$$

(3) 

When $A_3^{†}{C}_2{B}_0^{†}\ne 0$ or $A_0^{†}{C}_2{B}_3^{†}\ne 0$, the dual quaternion matrix Equation (1) does not admit a least-norm solution. Instead, it only has a solution with a minimum real part of the norm, while the dual part of the norm can take any real value. Accordingly, the solution $X=X_0+X_1\epsilon$ can be represented as

$$\begin{array}{cc}\hfill X_0& =A_0^{†}{C}_0{B}_0^{†}+A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill X_1& =A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0},\hfill \end{array}$$

(22)

where $W_i(i=1,2)$ are arbitrary matrices over $\mathbb{H}$ with appropriate dimensions. In this case,

$$\begin{array}{cc}\hfill \left|\right|X\left|\right|& =\left|\right|X_0{\left|\right|}_{min}+{\displaystyle \frac{a+K}{\left|\right|X_0{\left|\right|}_{min}}}\epsilon \hfill \\ \hfill & =\left|\right|A_0^{†}{C}_0{B}_0^{†}+A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†}\left|\right|+{\displaystyle \frac{a+K}{\left|\right|A_0^{†}{C}_0{B}_0^{†}+A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†}\left|\right|}}\epsilon ,\hfill \end{array}$$

where

$$a=\mathrm{Re}\left[\mathrm{tr}{(A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†})}^{*}\left(A_0^{†}{C}_0{B}_0^{†}\right)\right]$$

is a constant and

$$K=\mathrm{Re}\left[\mathrm{tr}(W_1^{*}{A}_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†}{W}_2^{*})\right]$$

can take any real value.

Proof. 

By Lemma 2, the general solution $X=X_0+X_1\epsilon$ to matrix Equation (1) can be expressed as follows:

$$\begin{array}{cc}\hfill X_0& =A_0^{†}{C}_0{B}_0^{†}+L_{A_0}{U}_1+U_2{R}_{B_0},\hfill \\ \hfill X_1& =A_0^{†}(C_2-A_2{U}_1{B}_0-A_0{U}_2{B}_2)B_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0},\hfill \\ \hfill U_1& =A_3^{†}{C}_{21}{B}_0^{†}+L_{A_3}{Q}_1+Q_2{R}_{B_0},\hfill \\ \hfill U_2& =A_0^{†}{C}_{22}{B}_3^{†}+L_{A_0}{Q}_3+Q_4{R}_{B_3},\hfill \end{array}$$

where $Q_i(i=\overline{1,4}),W_i(i=1,2)$ are arbitrary matrices over $\mathbb{H}$ with appropriate dimensions.

According to Proposition 1, we can find that the solution for $\parallel X\parallel$ can be translate into analyzing $\parallel X_0\parallel$, $\parallel X_1\parallel$, and Re[tr$X_1^{*}{X}_0$]. Then, by considering the total order relation for dual numbers, we understand that to minimize $\parallel X\parallel$, we first need to focus on minimizing the real part of $\parallel X\parallel$, i.e., $\parallel X_0\parallel$.

In view of Lemmas 3 and 4, we have

$$\begin{array}{cc}\hfill \left|\right|X_0{\left|\right|}^2& =\left|\right|A_0^{†}{C}_0{B}_0^{†}+L_{A_0}{U}_1+U_2{R}_{B_0}{\left|\right|}^2\hfill \\ \hfill & =\left|\right|A_0^{†}{C}_0{B}_0^{†}{\left|\right|}^2+\left|\right|L_{A_0}{U}_1+U_2{R}_{B_0}{\left|\right|}^2+2\mathrm{Re}\left[\mathrm{tr}{(L_{A_0}{U}_1+U_2{R}_{B_0})}^{*}{A}_0^{†}{C}_0{B}_0^{†}\right]\hfill \\ \hfill & =\left|\right|A_0^{†}{C}_0{B}_0^{†}{\left|\right|}^2+\left|\right|L_{A_0}{U}_1+U_2{R}_{B_0}{\left|\right|}^2+2\mathrm{Re}\left[\mathrm{tr}(U_1^{*}{L}_{A_0}{A}_0^{†}{C}_0{B}_0^{†}+\mathrm{tr}{R}_{B_0}{U}_2^{*}{A}_0^{†}{C}_0{B}_0^{†})\right]\hfill \\ \hfill & =\left|\right|A_0^{†}{C}_0{B}_0^{†}{\left|\right|}^2+\left|\right|L_{A_0}{U}_1+U_2{R}_{B_0}{\left|\right|}^2+2\mathrm{Re}\left[\mathrm{tr}{U}_2^{*}{A}_0^{†}{C}_0{B}_0^{†}{R}_{B_0}\right]\hfill \\ \hfill & =\left|\right|A_0^{†}{C}_0{B}_0^{†}{\left|\right|}^2+\left|\right|L_{A_0}{U}_1+U_2{R}_{B_0}{\left|\right|}^2,\hfill \end{array}$$

(23)

then,

$$\begin{array}{cc}\hfill \left|\right|L_{A_0}{U}_1+U_2{R}_{B_0}{\left|\right|}^2& =\left|\right|L_{A_0}(A_3^{†}{C}_{21}{B}_0^{†}+L_{A_3}{Q}_1+Q_2{R}_{B_0})\hfill \\ \hfill & +(A_0^{†}{C}_{22}{B}_3^{†}+L_{A_0}{Q}_3+Q_4{R}_{B_3})R_{B_0}{\left|\right|}^2\hfill \\ \hfill & =\left|\right|L_{A_0}{L}_{A_3}{Q}_1+L_{A_0}{Q}_2{R}_{B_0}+L_{A_0}{Q}_3{R}_{B_0}+Q_4{R}_{B_3}{R}_{B_0}{\left|\right|}^2\hfill \\ \hfill & +\left|\right|L_{A_0}{A}_3^{†}{C}_{21}{B}_0^{†}+A_0^{†}{C}_{22}{B}_3^{†}{R}_{B_0}{\left|\right|}^2+K_1\hfill \\ \hfill & =\left|\right|L_{A_0}{L}_{A_3}{Q}_1+L_{A_0}{Q}_2{R}_{B_0}+L_{A_0}{Q}_3{R}_{B_0}+Q_4{R}_{B_3}{R}_{B_0}{\left|\right|}^2\hfill \\ \hfill & +\left|\right|L_{A_0}{A}_3^{†}{C}_{21}{B}_0^{†}{\left|\right|}^2+\left|\right|A_0^{†}{C}_{22}{B}_3^{†}{R}_{B_0}{\left|\right|}^2+K_1\hfill \\ \hfill & +2\mathrm{Re}\left[\mathrm{tr}{\left(A_0^{†}{C}_{22}{B}_3^{†}{R}_{B_0}\right)}^{*}\left(L_{A_0}{A}_3^{†}{C}_{21}{B}_0^{†}\right)\right]\hfill \\ \hfill & =\left|\right|L_{A_0}{L}_{A_3}{Q}_1+L_{A_0}{Q}_2{R}_{B_0}+L_{A_0}{Q}_3{R}_{B_0}+Q_4{R}_{B_3}{R}_{B_0}{\left|\right|}^2\hfill \\ \hfill & +\left|\right|L_{A_0}{A}_3^{†}{R}_{A_0}{C}_2{B}_0^{†}{\left|\right|}^2+\left|\right|A_0^{†}{C}_2{L}_{B_0}{B}_3^{†}{R}_{B_0}{\left|\right|}^2+K_1\hfill \\ \hfill & +2\mathrm{Re}\left[\mathrm{tr}{\left(L_{A_0}{A}_0^{†}{C}_{22}{B}_3^{†}{R}_{B_0}\right)}^{*}\left(A_3^{†}{C}_{21}{B}_0^{†}\right)\right]\hfill \\ \hfill & =\left|\right|L_{A_0}{L}_{A_3}{Q}_1+L_{A_0}{Q}_2{R}_{B_0}+L_{A_0}{Q}_3{R}_{B_0}+Q_4{R}_{B_3}{R}_{B_0}{\left|\right|}^2\hfill \\ \hfill & +\left|\right|A_3^{†}{C}_2{B}_0^{†}{\left|\right|}^2+\left|\right|A_0^{†}{C}_2{B}_3^{†}{\left|\right|}^2+K_1,\hfill \end{array}$$

(24)

where

$$\begin{array}{c}\hfill K_1=2\mathrm{Re}\left[\mathrm{tr}{(L_{A_0}{L}_{A_3}{Q}_1+L_{A_0}{Q}_2{R}_{B_0}+L_{A_0}{Q}_3{R}_{B_0}+Q_4{R}_{B_3}{R}_{B_0})}^{*}(A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†})\right].\end{array}$$

According to Lemmas 3 and 4, we have

$$\begin{array}{cc}\hfill \mathrm{Re}[\mathrm{tr}{\left(L_{A_0}{L}_{A_3}{Q}_1\right)}^{*}(A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†})& =\mathrm{Re}\left[\mathrm{tr}(Q_1^{*}{L}_{A_3}{L}_{A_0}{A}_3^{†}{C}_2{B}_0^{†}+Q_1^{*}{L}_{A_3}{L}_{A_0}{A}_0^{†}{C}_2{B}_3^{†})\right]\hfill \\ \hfill & =\mathrm{Re}\left[\mathrm{tr}{Q}_1^{*}{L}_{A_3}{A}_3^{†}{C}_2{B}_0^{†}\right]\hfill \\ \hfill & =0,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{Re}[\mathrm{tr}{\left(L_{A_0}{Q}_2{R}_{B_0}\right)}^{*}(A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†})& =\mathrm{Re}\left[\mathrm{tr}(R_{B_0}{Q}_2^{*}{L}_{A_0}{A}_3^{†}{C}_2{B}_0^{†}+R_{B_0}{Q}_2^{*}{L}_{A_0}{A}_0^{†}{C}_2{B}_3^{†})\right]\hfill \\ \hfill & =\mathrm{Re}\left[\mathrm{tr}{Q}_2^{*}{L}_{A_0}{A}_3^{†}{C}_2{B}_0^{†}{R}_{B_0}\right]\hfill \\ \hfill & =0,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{Re}[\mathrm{tr}{\left(L_{A_0}{Q}_3{R}_{B_0}\right)}^{*}(A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†})& =\mathrm{Re}\left[\mathrm{tr}(R_{B_0}{Q}_3^{*}{L}_{A_0}{A}_3^{†}{C}_2{B}_0^{†}+R_{B_0}{Q}_3^{*}{L}_{A_0}{A}_0^{†}{C}_2{B}_3^{†})\right]\hfill \\ \hfill & =\mathrm{Re}\left[\mathrm{tr}{Q}_3^{*}{L}_{A_0}{A}_3^{†}{C}_2{B}_0^{†}{R}_{B_0}\right]\hfill \\ \hfill & =0,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{Re}[\mathrm{tr}{\left(Q_4{R}_{B_3}{R}_{B_0}\right)}^{*}(A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†})& =\mathrm{Re}\left[\mathrm{tr}(R_{B_0}{R}_{B_3}{Q}_4^{*}{A}_3^{†}{C}_2{B}_0^{†}+R_{B_0}{R}_{B_3}{Q}_4^{*}{A}_0^{†}{C}_2{B}_3^{†})\right]\hfill \\ \hfill & =\mathrm{Re}\left[\mathrm{tr}(R_{B_3}{Q}_4^{*}{A}_3^{†}{C}_2{B}_0^{†}{R}_{B_0}+Q_4^{*}{A}_0^{†}{C}_2{B}_3^{†}{R}_{B_0}{R}_{B_3})\right]\hfill \\ \hfill & =\mathrm{Re}\left[\mathrm{tr}{Q}_4^{*}{A}_0^{†}{C}_2{B}_3^{†}{R}_{B_3}\right]\hfill \\ \hfill & =0.\hfill \end{array}$$

Thus, $K_1=0$. It follows (23) and (24) that

$$\begin{array}{cc}\hfill \left|\right|X_0{\left|\right|}^2& =\left|\right|A_0^{†}{C}_0{B}_0^{†}{\left|\right|}^2+\left|\right|A_3^{†}{C}_2{B}_0^{†}{\left|\right|}^2+\left|\right|A_0^{†}{C}_2{B}_3^{†}{\left|\right|}^2\hfill \\ \hfill & +\left|\right|L_{A_0}{L}_{A_3}{Q}_1+L_{A_0}{Q}_2{R}_{B_0}+L_{A_0}{Q}_3{R}_{B_0}+Q_4{R}_{B_3}{R}_{B_0}{\left|\right|}^2\hfill \\ \hfill & \ge \left|\right|A_0^{†}{C}_0{B}_0^{†}{\left|\right|}^2+\left|\right|A_3^{†}{C}_2{B}_0^{†}{\left|\right|}^2+\left|\right|A_0^{†}{C}_2{B}_3^{†}{\left|\right|}^2,\hfill \end{array}$$

(25)

with equality if and only if

$$L_{A_0}{L}_{A_3}{Q}_1+L_{A_0}{Q}_2{R}_{B_0}+L_{A_0}{Q}_3{R}_{B_0}+Q_4{R}_{B_3}{R}_{B_0}=0,$$

(26)

i.e.,

$$X_0=A_0^{†}{C}_0{B}_0^{†}+A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†}.$$

(27)

Therefore,

$$\begin{array}{cc}\hfill \left|\right|X_0{\left|\right|}_{min}& =\left|\right|A_0^{†}{C}_0{B}_0^{†}+A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†}\left|\right|\hfill \\ \hfill & =\sqrt{\left|\right|A_0^{†}{C}_0{B}_0^{†}{\left|\right|}^2+\left|\right|A_3^{†}{C}_2{B}_0^{†}{\left|\right|}^2+\left|\right|A_0^{†}{C}_2{B}_3^{†}{\left|\right|}^2}.\hfill \end{array}$$

(28)

We now discuss $\left|\right|X\left|\right|$ under two different cases based on conditions (26) and (27).

(1)

If $A_0^{†}{C}_0{B}_0^{†}\ne 0$ or $A_3^{†}{C}_2{B}_0^{†}\ne 0$ or $A_0^{†}{C}_2{B}_3^{†}\ne 0$, by (25), we have that

$$\left|\right|X_0{\left|\right|}^2=\left|\right|A_0^{†}{C}_0{B}_0^{†}{\left|\right|}^2+\left|\right|A_3^{†}{C}_2{B}_0^{†}{\left|\right|}^2+\left|\right|A_0^{†}{C}_2{B}_3^{†}{\left|\right|}^2>0,$$

implying $X_0\ne 0$, i.e., X is appreciable. Consequently,

$$\left|\right|X\left|\right|=\left|\right|X_0\left|\right|+{\displaystyle \frac{\mathrm{Re}\left[\mathrm{tr}{X}_1^{*}{X}_0\right]}{\left|\right|X_0\left|\right|}}\epsilon .$$

Next, we turn our attention to the dual part of $\left|\right|X\left|\right|$. Having already examined $\left|\right|X_0\left|\right|$ mentioned above, we will now delve into the analysis of $\mathrm{Re}\left[\mathrm{tr}{X}_1^{*}{X}_0\right]$ in the following part.

It follows from (14), (15), (16), and (26) that

$$\begin{array}{cc}\hfill X_1& =A_0^{†}(C_2-A_2{U}_1{B}_0-A_0{U}_2{B}_2)B_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0}\hfill \\ \hfill & =A_0^{†}(C_2-A_2(A_3^{†}{C}_{21}{B}_0^{†}+L_{A_3}{Q}_1+Q_2{R}_{B_0})B_0\hfill \\ \hfill & -A_0(A_0^{†}{C}_{22}{B}_3^{†}+L_{A_0}{Q}_3+Q_4{R}_{B_3})B_2)B_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0}\hfill \\ \hfill & =A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_2(A_3^{†}{C}_{21}{B}_0^{†}+L_{A_3}{Q}_1)B_0{B}_0^{†}-A_0^{†}{A}_0(A_0^{†}{C}_{22}{B}_3^{†}+Q_4{R}_{B_3})B_2{B}_0^{†}\hfill \\ \hfill & +L_{A_0}{W}_1+W_2{R}_{B_0}\hfill \\ \hfill & =A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_2{A}_3^{†}{C}_{21}{B}_0^{†}-A_0^{†}{A}_2{L}_{A_3}{Q}_1{B}_0{B}_0^{†}-A_0^{†}{C}_{22}{B}_3^{†}{B}_2{B}_0^{†}\hfill \\ \hfill & -A_0^{†}{A}_0{Q}_4{R}_{B_3}{B}_2{B}_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0}\hfill \\ \hfill & =A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_2{A}_3^{†}{C}_{21}{B}_0^{†}-A_0^{†}{C}_{22}{B}_3^{†}{B}_2{B}_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0}\hfill \\ \hfill & -A_0^{†}{A}_1\left(L_{A_0}{L}_{A_3}{Q}_1\right)B_0{B}_0^{†}-A_0^{†}{A}_0\left(Q_4{R}_{B_3}{R}_{B_0}\right)B_1{B}_0^{†}\hfill \\ \hfill & =A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0}\hfill \\ \hfill & +A_0^{†}{A}_1(L_{A_0}{Q}_2{R}_{B_0}+L_{A_0}{Q}_3{R}_{B_0}+Q_4{R}_{B_3}{R}_{B_0})B_0{B}_0^{†}\hfill \\ \hfill & +A_0^{†}{A}_0(L_{A_0}{L}_{A_3}{Q}_1+L_{A_0}{Q}_2{R}_{B_0}+L_{A_0}{Q}_3{R}_{B_0})B_1{B}_0^{†}\hfill \\ \hfill & =A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0}.\hfill \end{array}$$

(29)

Thus,

$$\begin{array}{cc}\hfill \mathrm{Re}\left[\mathrm{tr}{X}_1^{*}{X}_0\right]& =\mathrm{Re}[\mathrm{tr}{(A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0})}^{*}\hfill \\ \hfill & (A_0^{†}{C}_0{B}_0^{†}+A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†})]\hfill \\ \hfill & =a+K,\hfill \end{array}$$

(30)

where,

$$\begin{array}{cc}\hfill a& =\mathrm{Re}\left[\mathrm{tr}{(A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†})}^{*}(A_0^{†}{C}_0{B}_0^{†}+A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†})\right]\hfill \\ \hfill & =\mathrm{Re}\left[\mathrm{tr}{(A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†})}^{*}\left(A_0^{†}{C}_0{B}_0^{†}\right)\right]\hfill \\ \hfill & +\mathrm{Re}\left[\mathrm{tr}{(A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†})}^{*}\left(L_{A_0}{A}_3^{†}{C}_2{B}_0^{†}\right)\right]\hfill \\ \hfill & +\mathrm{Re}\left[\mathrm{tr}{(A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†})}^{*}\left(A_0^{†}{C}_2{B}_3^{†}{R}_{B_0}\right)\right]\hfill \\ \hfill & =\mathrm{Re}\left[\mathrm{tr}{(A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†})}^{*}\left(A_0^{†}{C}_0{B}_0^{†}\right)\right]\hfill \\ \hfill & +\mathrm{Re}\left[\mathrm{tr}{(A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†})}^{*}{L}_{A_0}\left(A_3^{†}{C}_2{B}_0^{†}\right)\right]\hfill \\ \hfill & +\mathrm{Re}\left[\mathrm{tr}{R}_{B_0}{(A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†})}^{*}\left(A_0^{†}{C}_2{B}_3^{†}\right)\right]\hfill \\ \hfill & =\mathrm{Re}\left[\mathrm{tr}{(A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†})}^{*}\left(A_0^{†}{C}_0{B}_0^{†}\right)\right]\hfill \\ \hfill & +\mathrm{Re}\left[\mathrm{tr}{(C_2{B}_0^{†}-A_1{A}_3^{†}{C}_2{B}_0^{†}-C_2{B}_3^{†}{B}_1{B}_0^{†})}^{*}{\left(L_{A_0}{A}_0^{†}\right)}^{*}\left(A_3^{†}{C}_2{B}_0^{†}\right)\right]\hfill \\ \hfill & +\mathrm{Re}\left[\mathrm{tr}{\left(B_0^{†}{R}_{B_0}\right)}^{*}{(A_0^{†}{C}_2-A_0^{†}{A}_1{A}_3^{†}{C}_2-A_0^{†}{C}_2{B}_3^{†}{B}_1)}^{*}\left(A_0^{†}{C}_2{B}_3^{†}\right)\right]\hfill \\ \hfill & =\mathrm{Re}\left[\mathrm{tr}{(A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†})}^{*}\left(A_0^{†}{C}_0{B}_0^{†}\right)\right]\hfill \end{array}$$

(31)

is a constant, and

$$\begin{array}{cc}\hfill K& =\mathrm{Re}\left[\mathrm{tr}{(L_{A_0}{W}_1+W_2{R}_{B_0})}^{*}(A_0^{†}{C}_0{B}_0^{†}+A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†})\right]\hfill \\ \hfill & =\mathrm{Re}\left[\mathrm{tr}(W_1^{*}{L}_{A_0}{A}_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†}{R}_{B_0}{W}_2^{*})\right]\hfill \\ \hfill & =\mathrm{Re}\left[\mathrm{tr}(W_1^{*}{A}_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†}{W}_2^{*})\right].\hfill \end{array}$$

(32)

Next, we explore the situation of K in two cases.

(1.1)

If $A_3^{†}{C}_2{B}_0^{†}=A_0^{†}{C}_2{B}_3^{†}=0$, then substituting these into (31) and (32), we obtain

$$K=0,a=\mathrm{Re}\left[\mathrm{tr}{\left(A_0^{†}{C}_2{B}_0^{†}\right)}^{*}\left(A_0^{†}{C}_0{B}_0^{†}\right)\right].$$

According to (27) and (29), the least-norm solution to the dual quaternion matrix Equation (1) can be expressed as

$$\begin{array}{cc}\hfill X_0& =A_0^{†}{C}_0{B}_0^{†}+A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†}=A_0^{†}{C}_0{B}_0^{†},\hfill \\ \hfill X_1& =A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0}\hfill \\ \hfill & =A_0^{†}{C}_2{B}_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0},\hfill \end{array}$$

where $W_i(i=1,2)$ are arbitrary matrices over $\mathbb{H}$ with appropriate dimensions.

In this case, the least norm of the general solution can be expressed as

$$\begin{array}{cc}\hfill {\left|\right|X\left|\right|}_{min}& =\left|\right|X_0\left|\right|+{\displaystyle \frac{\mathrm{Re}\left[\mathrm{tr}{X}_1^{*}{X}_0\right]}{\left|\right|X_0\left|\right|}}\epsilon \hfill \\ \hfill & =\left|\right|A_0^{†}{C}_0{B}_0^{†}\left|\right|+{\displaystyle \frac{\mathrm{Re}\left[\mathrm{tr}{\left(A_0^{†}{C}_2{B}_0^{†}\right)}^{*}\left(A_0^{†}{C}_0{B}_0^{†}\right)\right]}{\left|\right|A_0^{†}{C}_0{B}_0^{†}\left|\right|}}\epsilon .\hfill \end{array}$$

(1.2)

If $A_3^{†}{C}_2{B}_0^{†}\ne 0$ or $A_0^{†}{C}_2{B}_3^{†}\ne 0$, it follows from (32) and Proposition 2 that K can take any real value. Thus, the dual part of $\left|\right|X\left|\right|$,

$${\displaystyle \frac{\mathrm{Re}\left[\mathrm{tr}{X}_1^{*}{X}_0\right]}{\left|\right|X_0\left|\right|}}={\displaystyle \frac{a+K}{\left|\right|A_0^{†}{C}_0{B}_0^{†}+A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†}\left|\right|}}$$

can take any real value. Therefore, the dual quaternion matrix Equation (1) does not have a least-norm solution in this case. However, it does have a solution with a minimum real part of the norm, while the dual part of the norm can take any real value. Accordingly, the solution $X=X_0+X_1\epsilon$ can be represented as

$$\begin{array}{cc}\hfill X_0& =A_0^{†}{C}_0{B}_0^{†}+A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†},\hfill \\ \hfill X_1& =A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0},\hfill \end{array}$$

where $W_i(i=1,2)$ are arbitrary matrices over $\mathbb{H}$ with appropriate dimensions. In such circumstances,

$$\begin{array}{cc}\hfill \left|\right|X\left|\right|& =\left|\right|X_0{\left|\right|}_{min}+{\displaystyle \frac{a+K}{\left|\right|X_0{\left|\right|}_{min}}}\epsilon \hfill \\ \hfill & =\left|\right|A_0^{†}{C}_0{B}_0^{†}+A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†}\left|\right|+{\displaystyle \frac{a+K}{\left|\right|A_0^{†}{C}_0{B}_0^{†}+A_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†}\left|\right|}}\epsilon ,\hfill \end{array}$$

where

$$a=\mathrm{Re}\left[\mathrm{tr}{(A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†})}^{*}\left(A_0^{†}{C}_0{B}_0^{†}\right)\right]$$

is a constant and

$$K=\mathrm{Re}\left[\mathrm{tr}(W_1^{*}{A}_3^{†}{C}_2{B}_0^{†}+A_0^{†}{C}_2{B}_3^{†}{W}_2^{*})\right]$$

can take any real value.

(2)

If $A_0^{†}{C}_0{B}_0^{†}=A_3^{†}{C}_2{B}_0^{†}=A_0^{†}{C}_2{B}_3^{†}=0$, by (25), we obtain

$$\left|\right|X_0{\left|\right|}^2=\left|\right|A_0^{†}{C}_0{B}_0^{†}{\left|\right|}^2+\left|\right|A_3^{†}{C}_2{B}_0^{†}{\left|\right|}^2+\left|\right|A_0^{†}{C}_2{B}_3^{†}{\left|\right|}^2=0.$$

Thus, $X_0=0$, i.e., X is infinitesimal. Therefore, $\left|\right|X\left|\right|=\left|\right|X_1\left|\right|\epsilon .$

According to (29), we have

$$\begin{array}{cc}\hfill X_1& =A_0^{†}{C}_2{B}_0^{†}-A_0^{†}{A}_1{A}_3^{†}{C}_2{B}_0^{†}-A_0^{†}{C}_2{B}_3^{†}{B}_1{B}_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0}\hfill \\ \hfill & =A_0^{†}(C_1-A_0{A}_0^{†}{C}_0{B}_0^{†}{B}_1-A_1{A}_0^{†}{C}_0{B}_0^{†}{B}_0)B_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0}\hfill \\ \hfill & =A_0^{†}{C}_1{B}_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0}.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill \left|\right|X_1{\left|\right|}^2& =\left|\right|A_0^{†}{C}_1{B}_0^{†}+L_{A_0}{W}_1+W_2{R}_{B_0}{\left|\right|}^2\hfill \\ \hfill & =\left|\right|A_0^{†}{C}_1{B}_0^{†}{\left|\right|}^2+\left|\right|L_{A_0}{W}_1+W_2{R}_{B_0}{\left|\right|}^2+2\mathrm{Re}\left[\mathrm{tr}{(L_{A_0}{W}_1+W_2{R}_{B_0})}^{*}\left(A_0^{†}{C}_1{B}_0^{†}\right)\right]\hfill \\ \hfill & =\left|\right|A_0^{†}{C}_1{B}_0^{†}{\left|\right|}^2+\left|\right|L_{A_0}{W}_1+W_2{R}_{B_0}{\left|\right|}^2+2\mathrm{Re}[\mathrm{tr}\left(W_1^{*}{L}_{A_0}{A}_0^{†}{C}_1{B}_0^{†}\right)+\mathrm{tr}\left(A_0^{†}{C}_1{B}_0^{†}{R}_{B_0}{W}_2^{*}\right)]\hfill \\ \hfill & =\left|\right|A_0^{†}{C}_1{B}_0^{†}{\left|\right|}^2+\left|\right|L_{A_0}{W}_1+W_2{R}_{B_0}{\left|\right|}^2\hfill \\ \hfill & \ge \left|\right|A_0^{†}{C}_1{B}_0^{†}{\left|\right|}^2,\hfill \end{array}$$

(33)

with equality if and only if

$$L_{A_0}{W}_1+W_2{R}_{B_0}=0,$$

(34)

i.e.,

$$X_1=A_0^{†}{C}_1{B}_0^{†}.$$

(35)

Therefore,

$$\begin{array}{c}\hfill \left|\right|X_1{\left|\right|}_{min}=\left|\right|A_0^{†}{C}_1{B}_0^{†}\left|\right|.\end{array}$$

(36)

Thus, the least-norm solution to the dual quaternion matrix Equation (1) can be expressed as

$$\begin{array}{cc}\hfill X_0& =0,\hfill \\ \hfill X_1& =A_0^{†}{C}_1{B}_0^{†}.\hfill \end{array}$$

In this case, the least norm of the general solution can be expressed as

$${\left|\right|X\left|\right|}_{min}=\left|\right|X_1\left|\right|\epsilon =\left|\right|A_0^{†}{C}_1{B}_0^{†}\left|\right|\epsilon .$$

□

As several applications of Theorem 1, we explore the least-norm solutions to the matrix equations $AX=B$ and $XC=D$ over the dual quaternion algebra.

Corollary 1. 

Consider $A=A_0+A_1\epsilon \in {\mathbb{DQ}}^{m\times n}$ and $B=B_0+B_1\epsilon \in {\mathbb{DQ}}^{m\times k}$, with $X=X_0+X_1\epsilon \in {\mathbb{DQ}}^{n\times k}$ as the unknown matrix. We assume that the dual quaternion matrix equation $AX=B$ is consistent. Set

$$\begin{array}{cc}\hfill & A_2=A_1{L}_{A_0},B_2=B_1-A_1{A}_0^{†}{B}_0,\hfill \\ \hfill & A_3=R_{A_0}{A}_2,B_3=R_{A_0}{B}_2.\hfill \end{array}$$

We discuss the least-norm solution to matrix Equation $AX=B$ in three cases:

(1) 

If $A_0^{†}{B}_0\ne 0$ and $A_3^{†}{B}_2=0$, then the least-norm solution $X=X_0+X_1\epsilon$ to matrix Equation $AX=B$ can be expressed as

$$\begin{array}{cc}\hfill X_0& =A_0^{†}{B}_0,\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill X_1& =A_0^{†}{B}_2+L_{A_0}{W}_1,\hfill \end{array}$$

(38)

where $W_1$ is an arbitrary matrix over $\mathbb{H}$ with appropriate dimensions.

In this case, the least norm of the general solution can be expressed as

$$\begin{array}{c}\hfill {\left|\right|X\left|\right|}_{min}=\left|\right|A_0^{†}{B}_0\left|\right|+{\displaystyle \frac{\mathrm{Re}\left[\mathrm{tr}{\left(A_0^{†}{B}_2\right)}^{*}\left(A_0^{†}{B}_0\right)\right]}{\left|\right|A_0^{†}{B}_0\left|\right|}}\epsilon .\end{array}$$

(2) 

Provided that $A_0^{†}{B}_0=A_3^{†}{B}_2=0$, matrix Equation $AX=B$ has a least-norm solution $X=X_0+X_1\epsilon$, which is given by

$$\begin{array}{cc}\hfill X_0& =0,\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill X_1& =A_0^{†}{B}_1.\hfill \end{array}$$

(40)

In such circumstances, the least norm of the general solution can be expressed as

$$\begin{array}{c}\hfill {\left|\right|X\left|\right|}_{min}=\left|\right|A_0^{†}{B}_1\left|\right|\epsilon .\end{array}$$

(3)

When $A_3^{†}{B}_2\ne 0$, the dual quaternion matrix equation $AX=B$ does not admit a least-norm solution. Instead, it only has a solution with a minimum real part of the norm, while the dual part of the norm can take any real value. Accordingly, the solution $X=X_0+X_1\epsilon$ can be represented as

$$\begin{array}{cc}\hfill X_0& =A_0^{†}{B}_0+A_3^{†}{B}_2,\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill X_1& =A_0^{†}{B}_2-A_0^{†}{A}_1{A}_3^{†}{B}_2+L_{A_0}{W}_1,\hfill \end{array}$$

(42)

where $W_i(i=1,2)$ are arbitrary matrices over $\mathbb{H}$ with appropriate dimensions. In this case,

$$\begin{array}{cc}\hfill \left|\right|X\left|\right|& =\left|\right|X_0{\left|\right|}_{min}+{\displaystyle \frac{a+K}{\left|\right|X_0{\left|\right|}_{min}}}\epsilon \hfill \\ \hfill & =\left|\right|A_0^{†}{B}_0+A_3^{†}{B}_2\left|\right|+{\displaystyle \frac{a+K}{\left|\right|A_0^{†}{B}_0+A_3^{†}{B}_2\left|\right|}}\epsilon ,\hfill \end{array}$$

where

$$a=\mathrm{Re}\left[\mathrm{tr}{(A_0^{†}{B}_2-A_0^{†}{A}_1{A}_3^{†}{B}_2)}^{*}{A}_0^{†}{B}_0\right]$$

is a constant and

$$K=\mathrm{Re}\left[\mathrm{tr}\left(W_1^{*}{A}_3^{†}{B}_2\right)\right]$$

can take any real value.

Corollary 2. 

Suppose $C=C_0+C_1\epsilon \in {\mathbb{DQ}}^{m\times n}$ and $D=D_0+D_1\epsilon \in {\mathbb{DQ}}^{l\times n}$ are known, where $X=X_0+X_1\epsilon \in {\mathbb{DQ}}^{l\times m}$ is the matrix to be determined. We consider the dual quaternion matrix equation $XC=D$ to be consistent. Define

$$\begin{array}{cc}\hfill & C_2=R_{C_0}{C}_1,D_2=D_1-D_0{C}_0^{†}{C}_1,\hfill \\ \hfill & C_3=C_2{L}_{C_0},D_3=D_2{L}_{C_0}.\hfill \end{array}$$

We investigate the least-norm solution to matrix Equation $XC=D$ in three cases:

(1) 

If $D_0{C}_0^{†}\ne 0$ and $D_2{C}_3^{†}=0$, then the least-norm solution $X=X_0+X_1\epsilon$ to matrix Equation $XC=D$ can be expressed as

$$\begin{array}{cc}\hfill X_0& =D_0{C}_0^{†},\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill X_1& =D_2{C}_0^{†}+W_1{R}_{C_0},\hfill \end{array}$$

(44)

where $W_1$ is an arbitrary matrix over $\mathbb{H}$ with appropriate dimensions.

In this case, the least norm of the general solution can be expressed as

$$\begin{array}{c}\hfill {\left|X\right||}_{min}=\left|\right|D_0{C}_0^{†}\left|\right|+{\displaystyle \frac{\mathrm{Re}\left[\mathrm{tr}{\left(D_2{C}_0^{†}\right)}^{*}\left(D_0{C}_0^{†}\right)\right]}{\left|\right|D_0{C}_0^{†}\left|\right|}}\epsilon .\end{array}$$

(2) 

Provided that $D_0{C}_0^{†}=D_2{C}_3^{†}=0$, matrix Equation $XC=D$ has a least-norm solution $X=X_0+X_1\epsilon$, which is given by

$$\begin{array}{cc}\hfill X_0& =0,\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill X_1& =D_1{C}_0^{†}.\hfill \end{array}$$

(46)

In such circumstances, the least norm of the general solution can be expressed as

$$\begin{array}{c}\hfill {\left|\right|X\left|\right|}_{min}=\left|\right|D_1{C}_0^{†}\left|\right|\epsilon .\end{array}$$

(3) 

When $D_2{C}_3^{†}\ne 0$, the dual quaternion matrix equation $XC=D$ does not admit a least-norm solution. Instead, it only has a solution with a minimum real part of the norm, while the dual part of the norm can take any real value. Accordingly, the solution $X=X_0+X_1\epsilon$ can be represented as

$$\begin{array}{cc}\hfill X_0& =D_0{C}_0^{†}+D_2{C}_3^{†},\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill X_1& =D_2{C}_0^{†}-D_2{C}_3^{†}{C}_1{C}_0^{†}+W_1{R}_{C_0},\hfill \end{array}$$

(48)

where $W_i(i=1,2)$ are arbitrary matrices over $\mathbb{H}$ with appropriate dimensions. In this case,

$$\begin{array}{cc}\hfill \left|\right|X\left|\right|& =\left|\right|X_0{\left|\right|}_{min}+{\displaystyle \frac{a+K}{\left|\right|X_0{\left|\right|}_{min}}}\epsilon \hfill \\ \hfill & =\left|\right|D_0{C}_0^{†}+D_2{C}_3^{†}\left|\right|+{\displaystyle \frac{a+K}{\left|\right|D_0{C}_0^{†}+D_2{C}_3^{†}\left|\right|}}\epsilon ,\hfill \end{array}$$

where

$$a=\mathrm{Re}\left[\mathrm{tr}{(D_2{C}_0^{†}-D_2{C}_3^{†}{C}_1{C}_0^{†})}^{*}\left(D_0{C}_0^{†}\right)\right]$$

is a constant and

$$K=\mathrm{Re}\left[\mathrm{tr}\left(D_2{C}_3^{†}{W}_1^{*}\right)\right]$$

can take any real value.

4. Numerical Examples

In this part, we will present two numerical examples to illustrate the conclusion in the previous section.

Example 1. 

Solving the least-norm solution to the dual quaternion matrix equation $AXB=C$, and providing the least norm of the general solution, where

$$\begin{array}{cc}\hfill A& =A_0+A_1\epsilon =\left[\begin{array}{cc}1+i& j-k\\ 2i+k& j\end{array}\right]+\left[\begin{array}{cc}2-i+k& k\\ i+j& 0\end{array}\right]\epsilon ,\hfill \\ \hfill B& =B_0+B_1\epsilon =\left[\begin{array}{cc}1+i+j& i+k\\ j-k& 0\end{array}\right]+\left[\begin{array}{cc}2i-k& i+j\\ 1& i\end{array}\right]\epsilon ,\hfill \\ \hfill C& =C_0+C_1\epsilon =\left[\begin{array}{cc}-3-i+3j+3k& -2-2i\\ -2-5i-j+4k& -4i-2k\end{array}\right]+\left[\begin{array}{cc}-6+7j+5k& -10+2i-3j-3k\\ -9-7i-3j+6k& -4-7i-2j-6k\end{array}\right]\epsilon .\hfill \end{array}$$

Through calculation, we have

$$\begin{array}{cc}\hfill & R_{A_0}{C}_0=0,C_0{L}_{B_0}=0,\hfill \\ \hfill & R_{A_3}{C}_{21}=0,R_{A_0}{C}_2{L}_{B_0}=0,C_{22}{L}_{B_3}=0.\hfill \end{array}$$

By Lemma 2, the matrix equation is consistent. On the other hand, we can identify that

$$A_0^{†}{C}_0{B}_0^{†}\ne 0\mathrm{and}{A}_3^{†}{C}_2{B}_0^{†}=A_0^{†}{C}_2{B}_3^{†}=0.$$

According to Theorem 1, the matrix equation has a least-norm solution, which can be expressed as

$$\begin{array}{c}\hfill X=X_0+X_1\epsilon =\left[\begin{array}{cc}i+k& 1\\ 0& 1+k\end{array}\right]+\left[\begin{array}{cc}2i+j& -1\\ k& 0\end{array}\right]\epsilon .\end{array}$$

In this case, the least norm of the general solution is $\left|\right|X\left|\right|=\sqrt{5}+{\displaystyle \frac{\sqrt{5}}{5}}\epsilon$.

Example 2. 

Finding the least-norm solution to the dual quaternion matrix equation $AXB=C$ and deriving the least norm of the general solution. Here, $A,B$, and C are given by

$$\begin{array}{cc}\hfill A& =A_0+A_1\epsilon =\left[\begin{array}{cc}1+i-j& k\\ i+j& i-k\end{array}\right]+\left[\begin{array}{cc}i-j& 0\\ i+2j& 1\end{array}\right]\epsilon ,\hfill \\ \hfill B& =B_0+B_1\epsilon =\left[\begin{array}{cc}1+i& 2-i\\ i+k& 1\end{array}\right]+\left[\begin{array}{cc}2-i& i+k\\ 0& j\end{array}\right]\epsilon ,\hfill \\ \hfill C& =C_0+C_1\epsilon =\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]+\left[\begin{array}{cc}1+j& 2-4i-2j+k\\ -4+2j& 1+6i+j+6k\end{array}\right]\epsilon .\hfill \end{array}$$

By means of calculation, we have

$$\begin{array}{cc}\hfill & R_{A_0}{C}_0=0,C_0{L}_{B_0}=0,\hfill \\ \hfill & R_{A_3}{C}_{21}=0,R_{A_0}{C}_2{L}_{B_0}=0,C_{22}{L}_{B_3}=0.\hfill \end{array}$$

In accordance with Lemma 2, the matrix equation is deemed to be consistent. Then, we can identify that

$$A_0^{†}{C}_0{B}_0^{†}=A_3^{†}{C}_2{B}_0^{†}=A_0^{†}{C}_2{B}_3^{†}=0.$$

According to Theorem 1, the matrix equation has a least-norm solution, which can be expressed as

$$\begin{array}{c}\hfill X=X_0+X_1\epsilon =\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]+\left[\begin{array}{cc}1& i+k\\ 2j-k& i\end{array}\right]\epsilon .\end{array}$$

In such circumstances, the least norm of the general solution is $\left|\right|X\left|\right|=3\epsilon$.

Example 3. 

Solving the least-norm solution to the dual quaternion matrix equation $AXB=C$, and providing the norm of the solution, where

$$\begin{array}{cc}\hfill A& =A_0+A_1\epsilon =\left[\begin{array}{cc}1+k& 2i-j\\ 0& 0\end{array}\right]+\left[\begin{array}{cc}i& 1+k\\ i+j& 2-i+j\end{array}\right]\epsilon ,\hfill \\ \hfill B& =B_0+B_1\epsilon =\left[\begin{array}{cc}1+i+j& i-j\\ j+k& 0\end{array}\right]+\left[\begin{array}{cc}i-k& 1+i\\ j& 1\end{array}\right]\epsilon ,\hfill \\ \hfill C& =C_0+C_1\epsilon =\left[\begin{array}{cc}2-2i+3j& -1+4i-k\\ 0& 0\end{array}\right]+\left[\begin{array}{cc}-4+3i+6j+2k& -3+7i+j-3k\\ -5+i+3j+3k& 2+2i-4k\end{array}\right]\epsilon .\hfill \end{array}$$

By calculation, we have

$$\begin{array}{cc}\hfill & R_{A_0}{C}_0=0,C_0{L}_{B_0}=0,\hfill \\ \hfill & R_{A_3}{C}_{21}=0,R_{A_0}{C}_2{L}_{B_0}=0,C_{22}{L}_{B_3}=0.\hfill \end{array}$$

It follows from Lemma 2 that the matrix equation is consistent. On the other hand, we can identify that

$$A_0^{†}{C}_0{B}_0^{†}\ne 0,A_3^{†}{C}_2{B}_0^{†}\ne 0\mathrm{and}{A}_0^{†}{C}_2{B}_3^{†}=0.$$

In view ofTheorem 1, the matrix equation does not have a least-norm solution and only has minimum real part of the norm solution. The solution $X=X_0+X_1\epsilon$ can be expressed as

$$\begin{array}{cc}\hfill X_0=& \left[\begin{array}{cc}2+j& 0\\ 1& i+k\end{array}\right],\hfill \\ \hfill X_1=& \left[\begin{array}{cc}0.2857+0.5714i& 0.1429+0.1429k\\ 0.2857-0.1429i+0.4286j-0.8571k& 0.1429i+0.2857j\end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{cc}0.7143& -0.1429i+0.4286j\\ 0.1429i-0.4286j& 0.2857\end{array}\right]W_1,\hfill \end{array}$$

where $W_1$ is an arbitrary matrix over $\mathbb{H}$ with appropriate dimensions. In this case, the norm of the solution is $\left|\right|X\left|\right|=2\sqrt{2}+{\displaystyle \frac{1+K}{2\sqrt{2}}}\epsilon$, where

$$K=\mathrm{Re}\left[\mathrm{tr}{W}_1^{*}\left[\begin{array}{cc}1.4286-0.1429i+1.1429j& 0.1429+0.4286i+0.1429j-0.4286k\\ 0.7143+0.2857i-0.8571j+0.1429k& 0.2857i+0.2857k\end{array}\right]\right].$$

5. Conclusions

In this paper, we first equivalently transform the expression of the norm of dual quaternion matrices as defined in reference [24]. Using this expression, we discuss the norm of the general solution to matrix Equation $AXB=C$ over the dual quaternion algebra when the equation is consistent and establish the conditions for the existence of the least-norm solution. A notable distinction emerges in the analysis of matrix Equation $AXB=C$ over the dual quaternion algebra. Crucially, this equation may not always admit a least-norm solution, even when it is consistent. This divergence stands in sharp contrast to the same equation $AXB=C$ over the quaternion algebra, which invariably possesses a least-norm solution whenever it is consistent. When the equation $AXB=C$ has a least-norm solution, we derive expressions for both the least-norm solution and the least norm of the general solution in two different cases. If no least-norm solution exists, we consider the minimum real part of the norm solution and establish expressions for the minimum real part of the norm solution and the norm of the solution. In this case, the dual part of norm of solution can take any real value. After that, we investigate the least-norm solution to the dual quaternion matrix equations $AX=B$ and $XC=D$ as an application. Finally, we present two numerical examples to illustrate the main results of this paper. In the future, we aim to investigate the least-norm solutions to more complex and general matrix equations over the dual quaternion algebra. Such solutions in quaternion systems have demonstrated broad utility, such as color image restoration (e.g., [26]). Moreover, unlike quaternion matrices, which encode a single color image, dual quaternion matrices can simultaneously represent two color images, suggesting that the least-norm solution over the dual quaternion algebra could offer significant advantages in color image restoration. Additionally, given the superiority of dual quaternions in simultaneously representing both the rotation and the translation of rigid bodies, they have found applications in the realms of 3D animated characters and coordinated robotic movements. We may be able to explore the applications of the least-norm solution of dual quaternion matrix equations in these two domains. Therefore, we will also investigate the potential applications of the least-norm solution within the dual quaternion algebra framework in future research.