The Dual Quaternion Matrix Equation AXB=C with Applications

1. Introduction

Let $\mathbb{R}$ denote the set of real numbers, ${\mathbb{H}}^{m\times n}$ stand for the space of all $m\times n$ matrices over the real quaternion algebra

$$\mathbb{H}=\{u_0+u_1\mathbf{i}+u_2\mathbf{j}+u_3\mathbf{k}|{\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=\mathbf{i}\mathbf{j}\mathbf{k}=-1,u_0,u_1,u_2,u_3\in \mathbb{R}\}.$$

The symbol of $r\left(A\right)$, $A^{\ast }$, I, and 0 are defined by the rank of a given quaternion matrix A, the conjugate transpose of A, identity matrix, zero matrix with appropriate sizes, respectively. The Moore-Penrose inverse of $A\in {\mathbb{H}}^{l\times k}$ is denoted as $A^{†}$, which is defined as the solution of $AYA=A,YAY=Y,{\left(AY\right)}^{\ast }=AY$ and ${\left(YA\right)}^{\ast }=YA.$ Moreover, let $L_A=I-A^{†}A$ and $R_A=I-A{A}^{†}$ represent two projectors along A.

Since Hamilton’s discovery of quaternions in 1843, quaternions and quaternion matrices have found a huge amount of practical applications in fields such as computer science, statistics, quantum physics, signal and color image processing, flight mechanics, aerospace technology, and so on (see, e.g., [1,2,3,4]).

Matrix equations have garnered significant attention in recent years and have become one of the main topics in matrix theory(see, e.g., [5,6,7,8,9]). For the classical matrix equation

$$\begin{array}{c}\hfill AXB=C,\end{array}$$

(1)

which has been studied by many authors. Ben-Israel and Greville [10] eatablished the necessary and sufficient condition for the solvability of (1). In 2003, Liao and Bai [11] investigated the least-squares solution of (1) over symmetric positive semidefinite matrices. Huang et al. [12] gave the skew-symmetric solution and the optimal approximate solution of (1). Peng [13] derived centro-symmetric solution of matrix equation (1). Deng et al. [14] studied the general expressions about the Hermitian solutions of (1). Xie and Wang [15] considered the reducible solution to (1) when it is solvable. As a special case of the matrix equation (1), the Hermitian solution X to the matrix equation

$$\begin{array}{c}\hfill AX{A}^{\ast }=B\end{array}$$

(2)

has attracted extensive attention(see, e.g., [16,17]). Baksalary[18] and Größ[19] studied the nonnegative definite and positive definite solutions to the matrix equation (2), respectively. For $\eta \in \{\mathbf{i},\mathbf{j},\mathbf{k}\}$, a quaternion matrix A is called $\eta$-Hermitian and $\eta$-skew-Hermitian if $A=A^{{\eta }^{\ast }}$ and $A=-A^{{\eta }^{\ast }}$, respectively, where $A^{{\eta }^{\ast }}=-\eta A^{\ast }\eta$[20]. The utilization of $\eta$-Hermitian matrices in linear modeling is extensively recognized[21]. Kyrchei[22] derived the explicit determinantal representation formulas of $\eta$-Hermitian and $\eta$-skew-Hermitian solutions to the quaternion matrix equation

$$AX{A}^{\eta \ast }=B.$$

It is well known that dual numbers and dual quaternions have wide applications in computer graphics, automatic differentiation, geometry, mechanics, rigid body motions and robotics(see, e.g., [23,24,25,26]). For the related definitions of dual numbers and dual quaternions, please see Section 2.

So far, there has been little information on matrix equation (1) over the dual quaternion algebra. Motivated by the work mentioned above, we in this paper aim to investigate the general solution of dual quaternion matrix equation (1) by using Moore–Penrose inverses and ranks of matrices. Since the $\varphi$-Hermitian serves as an extended form of both Hermiticity and $\eta$-Hermiticity over the quaternions [27], we also provide the definition of $\varphi$-Hermiticity over the dual quaternions. As an application, we establish the $\varphi$-Hermitian solution of a special dual quaternion matrix equation

$$\begin{array}{c}\hfill AX{A}^{\varphi }=C,C^{\varphi }=C.\end{array}$$

(3)

Further details regarding $\varphi$-Hermitian matrices will be illustrated in Section 2.

This paper is organized as follows. In Section 2, we provide an overview of essential definitions and lemmas that will be applied in the subsequent sections. In Section 3, we establish some necessary and sufficient conditions for solvability to the dual quaternion matrix equation (1) and consider some special cases of dual quaternion matrix equation (1). As an application, we investigate the $\varphi$-Hermitian solution of the dual quaternion matrix equation (3) in Section 4. In Section 5, we present a numerical example to illustrate the results of this paper. Finally a brief conclusion is provided in Section 6.

2. Preliminaries

In this section, we review some definitions of dual numbers, dual quaternions and related propositions. Moreover, we introduce the definitions of dual quaternion matrix and $\varphi$-Hermitian matrix which are fundamental for obtaining the main results.

Definition 1 

([28]). Suppose that $x_0,x_1\in \mathbb{R}$, we say x is a dual number if x has the form

$$\begin{array}{c}\hfill x=x_0+x_1ϵ,\end{array}$$

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

We call $x_0$ as the real part or the standard part of x, $x_1$ as the dual part or the infinitesimal part of x. The infinitesimal unit $ϵ$ is commutative in multiplication with real numbers, complex numbers, and quaternions. The set of dual numbers is denoted by

$$\mathbb{D}=\{x=x_0+x_1ϵ|{ϵ}^2=0,x_0,x_1\in \mathbb{R}\}.$$

Assume that $x=x_0+x_1ϵ,y=y_0+y_1ϵ\in \mathbb{D}$, we have $x=y$ if $x_0=y_0$ and $x_1=y_1$, regarding addition and multiplication, there is

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

Definition 2 

([28]). Let $z_0,z_1\in \mathbb{H}$. We say z is a dual quaternion if z has the form

$$\begin{array}{c}\hfill z=z_0+z_1ϵ,\end{array}$$

where ϵ is the infinitesimal unit, satisfying ${ϵ}^2=0$, $z_0,z_1$ as the real part and the dual part of z, respectively.

The collection of dual quaternions is denoted by

$$\mathbb{DQ}=\{z=z_0+z_1ϵ|{ϵ}^2=0,z_0,z_1\in \mathbb{H}\}.$$

Now, we introduce the definition of dual quaternion matrix. Let $X_0,X_1\in {\mathbb{H}}^{m\times n}$. X is said to be a dual quaternion matrix if X has the form $X=X_0+X_1ϵ$, the set of dual quaternion matrices is denoted by

$${\mathbb{DQ}}^{m\times n}=\{X=X_0+X_1ϵ|{ϵ}^2=0,X_0,X_1\in {\mathbb{H}}^{m\times n}\}.$$

The conjugate transpose of X is defined as $X^{\ast }=X_0^{\ast }+X_1^{\ast }ϵ$. For $Y=Y_0+Y_1ϵ\in {\mathbb{DQ}}^{m\times n}$, by analogy, we have $X=Y$ if $X_0=Y_0$ and $X_1=Y_1$, furthermore

$$\begin{array}{cc}\hfill & X+Y=X_0+Y_0+(X_1+Y_1)ϵ,\hfill \\ \hfill & XY=X_0{Y}_0+(X_0{Y}_1+X_1{Y}_0)ϵ.\hfill \end{array}$$

To facilitate our study on the $\varphi$-Hermitian, we will first review the concept of nonstandard involution over quaternions, and then proceed to generalize it to dual quaternions.

Definition 3 

([27]). A map $\varphi :\mathbb{H}\to \mathbb{H}$ is called an antiendomorphism, if $\varphi \left(pq\right)=\varphi \left(q\right)\varphi \left(p\right)$ and $\varphi (p+q)=\varphi \left(p\right)+\varphi \left(q\right)$ for all $p,q\in \mathbb{H}$. An antiendomorphism ϕ is called an involution if $\varphi \left(\varphi \right(p\left)\right)=p$ for every $p\in \mathbb{H}$.

Definition 4 

([27]). Under the basis $\left(1,i,j,k\right)$, an involution ϕ is called nonstandard if and only if ϕ can be expressed as a real matrix

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

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

Proposition 1 

([27]). Let $z\in \mathbb{H}$. Then every nonstandard involution ϕ of $\mathbb{H}$ has the form $\varphi \left(z\right)={\beta }^{-1}{z}^{\ast }\beta$ for some $\beta \in \mathbb{H}$ with ${\beta }^2=-1$.

Definition 5 

([27]). For a nonstandard involution ϕ, a quaternion matrix Z is said to be ϕ-Hermitian if $Z=Z^{\varphi }$, where $Z^{\varphi }$ is obtained by applying ϕ entrywise to the transposed matrix of Z. In fact, $Z^{\varphi }={\beta }^{-1}{Z}^{\ast }\beta$.

Proposition 2. 

Let $Z\in {\mathbb{H}}^{m\times n},\beta \in \mathbb{H}$ and ${\beta }^2=-1$. Then

$\left(1\right)$${\left(Z^{\varphi }\right)}^{†}={\left(Z^{†}\right)}^{\varphi }$,

$\left(2\right)$${\left(L_Z\right)}^{\varphi }=R_{Z^{\varphi }}$,

$\left(3\right)$${\left(R_Z\right)}^{\varphi }=L_{Z^{\varphi }}$.

Proof. 

It is obvious that ${\beta }^{-1}=-\beta ,{\beta }^{\ast }=-\beta$, by the definition of Moore-Penrose inverse, we obtain

$$\begin{array}{cc}\hfill Z^{\varphi }{\left(Z^{†}\right)}^{\varphi }{Z}^{\varphi }& =(-\beta Z^{\ast }\beta )[-\beta {\left(Z^{†}\right)}^{\ast }\beta ](-\beta Z^{\ast }B)\hfill \\ \hfill & =-\beta Z^{\ast }{\left(Z^{†}\right)}^{\ast }{Z}^{\ast }\beta =-\beta {\left(Z{Z}^{†}Z\right)}^{\ast }\beta \hfill \\ \hfill & =-\beta Z^{\ast }\beta =Z^{\varphi },\hfill \\ \hfill {\left(Z^{†}\right)}^{\varphi }{Z}^{\varphi }{\left(Z^{†}\right)}^{\varphi }& =[-\beta {\left(Z^{†}\right)}^{\ast }\beta ](-\beta Z^{\ast }\beta )[-\beta {\left(Z^{†}\right)}^{\ast }\beta ]\hfill \\ \hfill & =-\beta {\left(Z^{†}\right)}^{\ast }{Z}^{\ast }{\left(Z^{†}\right)}^{\ast }\beta =-\beta {\left(Z^{†}Z{Z}^{†}\right)}^{\ast }\beta \hfill \\ \hfill & =-\beta {\left(Z^{†}\right)}^{\ast }\beta ={\left(Z^{†}\right)}^{\varphi },\hfill \\ \hfill {}^{\varphi }\left[Z\right(Z^{†})^{\varphi }{]}^{\ast }& ={\left[(-\beta Z^{\ast }\beta )(-\beta {\left(Z^{†}\right)}^{\ast }\beta )\right]}^{\ast }\hfill \\ \hfill & ={[-\beta Z^{\ast }{\left(Z^{†}\right)}^{\ast }\beta ]}^{\ast }={(-\beta Z^{†}Z\beta )}^{\ast }\hfill \\ \hfill & =-\beta {\left(Z^{†}Z\right)}^{\ast }\beta =(-\beta Z^{\ast }\beta )[-\beta {\left(Z^{†}\right)}^{\ast }\beta ]\hfill \\ \hfill & =Z^{\varphi }{\left(Z^{†}\right)}^{\varphi },\hfill \\ \hfill [(Z^{†}{{)}^{\varphi }{Z}^{\varphi }]}^{\ast }^{†}& ={[-\beta {\left(Z^{†}\right)}^{\ast }\beta (-\beta Z^{\ast }\beta )]}^{\ast }\hfill \\ \hfill & ={[-\beta {\left(Z^{†}\right)}^{\ast }{Z}^{\ast }\beta ]}^{\ast }={(-\beta Z{Z}^{†}\beta )}^{\ast }\hfill \\ \hfill & =-\beta {\left(Z{Z}^{†}\right)}^{\ast }\beta =[-\beta {\left(Z^{†}\right)}^{\ast }\beta ](-\beta Z^{\ast }\beta )\hfill \\ \hfill & ={\left(Z^{†}\right)}^{\varphi }{Z}^{\varphi }.\hfill \end{array}$$

(4)

Based on this, we can deduce that ${\left(Z^{†}\right)}^{\varphi }$ is the Moore-Penrose inverse of $Z^{\varphi }$.

For $\left(2\right)$ , we have

$${\left(L_Z\right)}^{\varphi }={(I-Z^{†}Z)}^{\varphi }=I-Z^{\varphi }{\left(Z^{†}\right)}^{\varphi }=I-Z^{\varphi }{\left(Z^{\varphi }\right)}^{†}=R_{Z^{\varphi }}.$$

In a similar vein to $\left(2\right)$, we can offer a demonstration for $\left(3\right)$, therefore, we omit it here. □

By analogy, we propose the definition of $\varphi$-Hermiticity with respect to dual quaternion matrix, where $\varphi$ is a nonstandard involution.

Definition 6. 

For $X=X_0+X_1ϵ\in {\mathbb{DQ}}^{m\times n},$X is called ϕ-Hermitian matrix if $X=X^{\varphi }$, where

$$X^{\varphi }:={\beta }^{-1}{X}^{\ast }\beta ={\beta }^{-1}{X}_0^{\ast }\beta +{\beta }^{-1}{X}_1^{\ast }\beta ϵ=X_0^{\varphi }+X_1^{\varphi }ϵ,$$

with $\beta \in \mathbb{H}$ and ${\beta }^2=-1$.

Proposition 3. 

Let $X,Y\in {\mathbb{DQ}}^{n\times n}$. Then

$\left(1\right)$${(X+Y)}^{\varphi }=X^{\varphi }+Y^{\varphi }$,

$\left(2\right)$${\left(XY\right)}^{\varphi }=Y^{\varphi }{X}^{\varphi }$,

$\left(3\right)$${\left(X^{\varphi }\right)}^{\varphi }=X$.

Proof. 

By the algebraic properties of $\varphi$, we have

$$\begin{array}{cc}\hfill {(X+Y)}^{\varphi }& ={[(X_0+Y_0)+(X_1+Y_1)ϵ]}^{\varphi }\hfill \\ \hfill & ={(X_0+Y_0)}^{\varphi }+{(X_1+Y_1)}^{\varphi }ϵ\hfill \\ \hfill & =X_0^{\varphi }+Y_0^{\varphi }+X_1^{\varphi }ϵ+Y_1^{\varphi }ϵ\hfill \\ \hfill & =X_0^{\varphi }+X_1^{\varphi }ϵ+Y_0^{\varphi }+Y_1^{\varphi }ϵ\hfill \\ \hfill & =X^{\varphi }+Y^{\varphi }.\hfill \end{array}$$

In relation to $\left(2\right)$, we obtain

$$\begin{array}{cc}\hfill {\left(XY\right)}^{\varphi }& ={[X_0{Y}_0+(X_0{Y}_1+X_1{Y}_0)ϵ]}^{\varphi }\hfill \\ \hfill & ={\left(X_0{Y}_0\right)}^{\varphi }+{(X_0{Y}_1+X_1{Y}_0)}^{\varphi }ϵ\hfill \\ \hfill & =Y_0^{\varphi }{X}_0^{\varphi }+Y_1^{\varphi }{X}_0^{\varphi }ϵ+Y_0^{\varphi }{X}_1^{\varphi }ϵ\hfill \\ \hfill & =(Y_0^{\varphi }+Y_1^{\varphi }ϵ)(X_0^{\varphi }+X_1^{\varphi }ϵ)\hfill \\ \hfill & =Y^{\varphi }{X}^{\varphi }.\hfill \end{array}$$

In terms of $\left(3\right)$, we have

$$\begin{array}{cc}\hfill {\left(X^{\varphi }\right)}^{\varphi }& ={\left[{(X_0+X_1ϵ)}^{\varphi }\right]}^{\varphi }\hfill \\ \hfill & ={(X_0^{\varphi }+X_1^{\varphi }ϵ)}^{\varphi }\hfill \\ \hfill & =X_0+X_1ϵ\hfill \\ \hfill & =X.\hfill \end{array}$$

□

Now, we provide a few lemmas which are basic tools for getting the key outcomes.

Lemma 1 

([29]). Suppose that A, B and C are provided for matrices with the adequate dimensions over $\mathbb{H}$, then the quaternion matrix equation (1) is consistent if and only if

$$R_{A}C=0,C{L}_B=0.$$

In this case, the general solution can be expressed as

$$\begin{array}{c}\hfill X=A^{†}C{B}^{†}+L_{A}U+V{R}_B,\end{array}$$

where $U,V$ are any matrices over $\mathbb{H}$ with appropriate dimensions.

Lemma 2 

([15]). Let $A_1,A_2,B_1,B_2$ and $C_1$ be given with appropriate sizes. Set

$$A=R_{A_1}C,B=B_1{L}_{B_2},M=R_{A_1}{A}_2,C_1=C{L}_{B_2}.$$

Then the following descriptions are equivalent:

$\left(1\right)$ The quaternion matrix equation

$$A_1{X}_1{B}_1+A_1{X}_2{B}_2+A_2{X}_3{B}_2=C$$

(5)

is consistent.

$\left(2\right)$$R_{M}A=0,R_{A_1}C{L}_{B_2}=0,C_1{L}_B=0.$

$\left(3\right)$

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccc}{A}_1& A_2& C\end{array}\right)=r\left(\begin{array}{cc}{A}_1& A_2\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{cc}{B}_2& 0\\ C& A_1\end{array}\right)=r\left(\begin{array}{c}{B}_2\end{array}\right)+r\left(\begin{array}{c}{A}_1\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{c}{C}_1\\ B_1\\ B_2\end{array}\right)=r\left(\begin{array}{c}{B}_1\\ B_2\end{array}\right).\hfill \end{array}$$

In this case, the general solution to (5) can be expressed as follows:

$$\begin{array}{cc}\hfill & X_1=A_1^{†}{C}_1{B}^{†}+L_{A_1}{V}_1+V_2{R}_B,\hfill \\ \hfill & X_2=A_1^{†}(C-A_1{X}_1{B}_1-A_2{X}_3{B}_2)B_2^{†}+T_1{R}_{B_2}+L_{A_1}{T}_2,\hfill \\ \hfill & X_3=M^{†}A{B}_2^{†}+L_M{U}_1+U_2{R}_{B_2},\hfill \end{array}$$

where $U_1,U_2,V_1,V_2,T_1$, and $T_2$ are arbitrary matrices over $\mathbb{H}$ with appropriate sizes.

The following lemma, originally derived by Marsaglia and Styan[30], which can be extended to $\mathbb{H}$.

Lemma 3. 

Assume that $A\in {\mathbb{H}}^{m\times n},$$B\in {\mathbb{H}}^{m\times k},C\in {\mathbb{H}}^{l\times n}$, $D\in {\mathbb{H}}^{j\times k}$ and $E\in {\mathbb{H}}^{l\times i}$, then we have the following rank equality:

$$r\left(\begin{array}{cc}A& B{L}_D\\ R_{E}C& 0\end{array}\right)=r\left(\begin{array}{ccc}A& B& 0\\ C& 0& E\\ 0& D& 0\end{array}\right)-r\left(D\right)-r\left(E\right).$$

3. The solution of the matrix equation (1)

In this section, we establish the necessary and sufficient conditions for the solvability of the dual quaternion matrix equation (1) and provide the expressions of its general solution. Additionally, we investigate some special cases of the dual quaternion matrix equation (1).

Theorem 1. 

Let $A=A_0+A_1ϵ\in {\mathbb{DQ}}^{m\times n}$, $B=B_0+B_1ϵ\in {\mathbb{DQ}}^{k\times l}$, $C=C_0+C_1ϵ\in {\mathbb{DQ}}^{m\times l}$ be known. Put

$$A_2=A_1{L}_{A_0},B_2=R_{B_0}{B}_1,C_{11}=A_0{A}_0^{†}{C}_0{B}_0^{†}{B}_1,$$

(6)

$$C_{22}=A_1{A}_0^{†}{C}_0{B}_0^{†}{B}_0,C_2=C_1-C_{11}-C_{22},$$

(7)

$$M=R_{A_0}{A}_2,N=R_{A_0}{C}_2,E=B_2{L}_{B_0},F=C_2{L}_{B_0}.$$

(8)

Then the following statements are equivalent:

$\left(1\right)$ The dual quaternion matrix equation (1) is consistent.

$\left(2\right)$

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

(9)

$$\begin{array}{cc}\hfill & R_{M}N=0,R_{A_0}{C}_2{L}_{B_0}=0,F{L}_E=0.\hfill \end{array}$$

(10)

$\left(3\right)$

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cc}{A}_0& C_0\end{array}\right)=r\left(\begin{array}{c}{A}_0\end{array}\right),r\left(\begin{array}{c}{B}_0\\ C_0\end{array}\right)=r\left(\begin{array}{c}{B}_0\end{array}\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccc}{A}_1& A_0& C_1\\ A_0& 0& C_0\end{array}\right)=r\left(\begin{array}{cc}{A}_1& A_0\\ A_0& 0\end{array}\right),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cc}{C}_1& A_0\\ B_0& 0\end{array}\right)=r\left(\begin{array}{c}{A}_0\end{array}\right)+r\left(\begin{array}{c}{B}_0\end{array}\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cc}{B}_1& B_0\\ B_0& 0\\ C_1& C_0\end{array}\right)=r\left(\begin{array}{cc}{B}_1& B_0\\ B_0& 0\end{array}\right).\hfill \end{array}$$

(14)

In this case, the general solution X of dual quaternion matrix equation (1) can be expressed as $X=X_0+X_1ϵ$, where

$$\begin{array}{cc}\hfill & X_0=A_0^{†}{C}_0{B}_0^{†}+L_{A_0}U+V{R}_{B_0},\hfill \\ \hfill & X_1=A_0^{†}(C_2-A_{0}V{B}_2-A_{2}U{B}_0)B_0^{†}+W_1{R}_{B_0}+L_{A_0}{W}_2,\hfill \\ \hfill & U=M^{†}N{B}_0^{†}+L_M{Q}_1+Q_2{R}_{B_0},\hfill \\ \hfill & V=A_0^{†}F{E}^{†}+L_{A_0}{Q}_3+Q_4{R}_E,\hfill \end{array}$$

(15)

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

Proof.$\left(1\right)\iff \left(2\right)$: Suppose the dual quaternion matrix equation (1) is solvable and its solution $X\in {\mathbb{DQ}}^{m\times n}$ which can be expressed as

$$X=X_0+X_1ϵ,$$

(16)

substituting (16) into (1), by the definition of equality of dual quaternion matrices, we can get that dual quaternion matrix equation (1) is equivalent to the system of quaternion matrix equations

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & A_0{X}_0{B}_0=C_0,\hfill \\ \hfill & A_0{X}_0{B}_1+A_0{X}_1{B}_0+A_1{X}_0{B}_0=C_1.\hfill \end{array}\right.\end{array}$$

(17)

The structure of the proof goes as follows. We first prove that $\left(1\right)\iff \left(2\right)$ and illustrate the general solution of (17) have the form of (15), then we prove $\left(2\right)\iff \left(3\right)$.

We divide the system (17) into the following:

$$\begin{array}{c}\hfill A_0{X}_0{B}_0=C_0,\end{array}$$

(18)

and

$$\begin{array}{c}\hfill A_0{X}_0{B}_1+A_0{X}_1{B}_0+A_1{X}_0{B}_0=C_1.\end{array}$$

(19)

By Lemma 1, we obtain (18) is consistent if and only if

$$R_{A_0}{C}_0,C_0{L}_{B_0}=0.$$

In this case, the general solution of (18) can be written as

$$\begin{array}{c}\hfill X_0=A_0^{†}{C}_0{B}_0^{†}+L_{A_0}U+V{R}_{B_0},\end{array}$$

(20)

where $U,V$ are any matrices over $\mathbb{H}$ with appropriate sizes.

By substituting equation (20) into equation (19) gives

$$A_{0}V{B}_2+A_0{X}_1{B}_0+A_{2}U{B}_0=C_2,$$

(21)

where $A_2,B_2,$ and $C_2$ are defined by (6)-(7). Using Lemma 2 to (21), we know that matrix equation (21) is solvable if and only if

$$\begin{array}{c}\hfill R_{M}N=0,R_{A_0}{C}_2{L}_{B_0}=0,F{L}_E=0,\end{array}$$

where $M,N,E$ and F are given by (8). In this case, the general solution of (21) can be expressed as

$$\begin{array}{cc}\hfill & X_1=A_0^{†}(C_2-A_{0}V{B}_2-A_{2}U{B}_0)B_0^{†}+W_1{R}_{B_0}+L_{A_0}{W}_2,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & U=M^{†}N{B}_0^{†}+L_M{Q}_1+Q_2{R}_{B_0},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & V=A_0^{†}F{E}^{†}+L_{A_0}{Q}_3+Q_4{R}_E,\hfill \end{array}$$

(24)

where $Q_1,Q_2$, $Q_3$, $Q_4$, $W_1$ and $W_2$ are any matrices with the suitable dimensions over $\mathbb{H}$. To sum up, we have shown that the matrix equation (1) has a dual solution $X\in {\mathbb{DQ}}^{m\times n}$ if and only if $\left(2\right)$ holds.

$\left(2\right)\iff \left(3\right)$: We devide it into two parts to prove its equivalence.

$Part1$. In this part, we prove that (9) hold if and only if (11) hold. According to Lemma 3, it is easy to verify that $\left(9\right)\iff \left(11\right)$.

$Part2$. In this part, we turn to prove that $\left(10\right)\iff \left(12\right)-\left(14\right)$. Let $X_0=A_0^{†}{C}_0{B}_0^{†}.$ Then it is easy to verify that $X_0$ is a particular solution to the matrix equation $A_0{X}_0{B}_0=C_0.$ By Lemma 3 and block elementary operations, we obtain

$$\begin{array}{cc}\hfill R_{M}N=0& \iff r\left(R_{M}N\right)=0\iff r\left(\begin{array}{cc}{R}_{A_0}{A}_2& R_{A_0}{C}_2\end{array}\right)=r\left(R_{A_0}{A}_2\right),\hfill \\ \hfill & \iff r\left(\begin{array}{ccc}{A}_0& A_2& C_2\end{array}\right)=r\left(\begin{array}{cc}{A}_0& A_2\end{array}\right),\hfill \\ \hfill & \iff r\left(\begin{array}{ccc}{A}_0& A_1{L}_{A_0}& C_2\end{array}\right)=r\left(\begin{array}{cc}{A}_0& A_1{L}_{A_0}\end{array}\right),\hfill \\ \hfill & \iff r\left(\begin{array}{ccc}{A}_1& A_0& C_2\\ A_0& 0& 0\end{array}\right)=r\left(\begin{array}{cc}{A}_1& A_0\\ A_0& 0\end{array}\right),\hfill \\ \hfill & \iff r\left(\begin{array}{ccc}{A}_1& A_0& C_1-A_0{A}_0^{†}{C}_0{B}_0^{†}{B}_1-A_1{A}_0^{†}{C}_0{B}_0^{†}{B}_0\\ A_0& 0& 0\end{array}\right)=r\left(\begin{array}{cc}{A}_1& A_0\\ A_0& 0\end{array}\right),\hfill \\ \hfill & \iff r\left(\begin{array}{ccc}{A}_1& A_0& C_1\\ A_0& 0& C_0\end{array}\right)=r\left(\begin{array}{cc}{A}_1& A_0\\ A_0& 0\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill R_{A_0}{C}_2{L}_{B_0}=0& \iff r\left(R_{A_0}{C}_2{L}_{B_0}\right)=0\iff r\left(\begin{array}{cc}{C}_2& A_0\\ B_0& 0\end{array}\right)=r\left(A_0\right)+r\left(B_0\right),\hfill \\ \hfill & \iff r\left(\begin{array}{cc}{C}_1-C_{11}-C_{22}& A_0\\ B_0& 0\end{array}\right)=r\left(A_0\right)+r\left(B_0\right),\hfill \\ \hfill & \iff r\left(\begin{array}{cc}{C}_1-A_0{A}_0^{†}{C}_0{B}_0^{†}{B}_1-A_1{A}_0^{†}{C}_0{B}_0^{†}{B}_0& A_0\\ B_0& 0\end{array}\right)=r\left(A_0\right)+r\left(B_0\right),\hfill \\ \hfill & \iff r\left(\begin{array}{cc}{C}_1& A_0\\ B_0& 0\end{array}\right)=r\left(A_0\right)+r\left(B_0\right),\hfill \\ \hfill F{L}_E=0& \iff r\left(F{L}_E\right)=0\iff r\left(\begin{array}{c}E\\ F\end{array}\right)=r\left(E\right)\iff r\left(\begin{array}{c}{B}_2{L}_{B_0}\\ C_2{L}_{B_0}\end{array}\right)=r\left(B_2{L}_{B_0}\right),\hfill \\ \hfill & \iff r\left(\begin{array}{c}{B}_0\\ B_2\\ C_2\end{array}\right)=r\left(\begin{array}{c}{B}_0\\ B_2\end{array}\right)\iff r\left(\begin{array}{c}{B}_0\\ R_{B_0}{B}_1\\ C_2\end{array}\right)=r\left(\begin{array}{c}{B}_0\\ R_{B_0}{B}_1\end{array}\right),\hfill \\ \hfill & \iff r\left(\begin{array}{cc}{B}_1& B_0\\ B_0& 0\\ C_2& 0\end{array}\right)=r\left(\begin{array}{cc}{B}_1& B_0\\ B_0& 0\end{array}\right),\hfill \\ \hfill & \iff r\left(\begin{array}{cc}{B}_1& B_0\\ B_0& 0\\ C_1-A_0{A}_0^{†}{C}_0{B}_0^{†}{B}_1-A_1{A}_0^{†}{C}_0{B}_0^{†}{B}_0& 0\end{array}\right)=r\left(\begin{array}{cc}{B}_1& B_0\\ B_0& 0\end{array}\right),\hfill \\ \hfill & \iff r\left(\begin{array}{cc}{B}_1& B_0\\ B_0& 0\\ C_1& C_0\end{array}\right)=r\left(\begin{array}{cc}{B}_1& B_0\\ B_0& 0\end{array}\right).\hfill \end{array}$$

□

Now, we consider some special cases of dual quarernion matrix equation(1).

Corollary 1 

([31]). Assume that $A=A_0+A_1ϵ\in {\mathbb{DQ}}^{m\times n}$, $C=C_0+C_1ϵ\in {\mathbb{DQ}}^{m\times l}$ are given. Put

$$A_2=A_1{L}_{A_0},C_{22}=A_1{A}_0^{†}{C}_0,C_2=C_1-C_{22},M=R_{A_0}{A}_2,N=R_{A_0}{C}_2.$$

(25)

Then the following statements are equivalent:

$\left(1\right)$ The dual quaternion matrix equation $AX=C$ is consistent.

$\left(2\right)$

$$R_{A_0}{C}_0=0,R_{M}N=0.$$

(26)

$\left(3\right)$

$$r\left(\begin{array}{cc}{A}_0& C_0\end{array}\right)=r\left(\begin{array}{c}{A}_0\end{array}\right),r\left(\begin{array}{ccc}{A}_1& A_0& C_1\\ A_0& 0& C_0\end{array}\right)=r\left(\begin{array}{cc}{A}_1& A_0\\ A_0& 0\end{array}\right).$$

(27)

In this case, the general solution X of dual quaternion matrix equation $AX=C$ can be expressed as $X=X_0+X_1ϵ$, where

$$\begin{array}{cc}\hfill & X_0=A_0^{†}{C}_0+L_{A_0}U,\hfill \\ \hfill & X_1=A_0^{†}(C_2-A_{2}U)+L_{A_0}{W}_1,\hfill \\ \hfill & U=M^{†}N+L_M{W}_2,\hfill \end{array}$$

(28)

and $W_1,W_2$ are arbitrary matrices over $\mathbb{H}$ with appropriate dimensions.

Corollary 2 

([31]). Let $B=B_0+B_1ϵ\in {\mathbb{DQ}}^{k\times l}$, $C=C_0+C_1ϵ\in {\mathbb{DQ}}^{m\times l}$ be known. Denote

$$B_2=R_{B_0}{B}_1,C_{11}=C_0{B}_0^{†}{B}_1,C_2=C_1-C_{11},E=B_2{L}_{B_0},F=C_2{L}_{B_0}.$$

(29)

Then the following statements are equivalent:

$\left(1\right)$ The dual quaternion matrix equation $XB=C$ is consistent.

$\left(2\right)$

$$C_0{L}_{B_0}=0,F{L}_E=0.$$

(30)

$\left(3\right)$

$$r\left(\begin{array}{c}{B}_0\\ C_0\end{array}\right)=r\left(\begin{array}{c}{B}_0\end{array}\right),r\left(\begin{array}{cc}{B}_1& B_0\\ B_0& 0\\ C_1& C_0\end{array}\right)=r\left(\begin{array}{cc}{B}_1& B_0\\ B_0& 0\end{array}\right).$$

(31)

In this case, the general solution can be expressed as $X=X_0+X_1ϵ$, where

$$\begin{array}{cc}\hfill & X_0=C_0{B}_0^{†}+V{R}_{B_0},\hfill \\ \hfill & X_1=(C_2-V{B}_2)B_0^{†}+W_1{R}_{B_0},\hfill \\ \hfill & V=F{E}^{†}+W_2{R}_E,\hfill \end{array}$$

(32)

and $W_1,W_2$ are arbitrary matrices over $\mathbb{H}$ with appropriate dimensions.

4. Applications

As an application of Theorem 1, we can investigate the dual quaternion matrix equation (3).

Theorem 2. 

Suppose that $A=A_0+A_1ϵ\in {\mathbb{DQ}}^{m\times n},C=C_0+C_1ϵ=C^{\varphi }\in {\mathbb{DQ}}^{m\times m}$ are given, denote

$$\begin{array}{cc}\hfill & B_2=R_{A_0^{\varphi }}{A}_1^{\varphi },C_{11}=A_0{A}_0^{†}{C}_0{\left(A_0^{\varphi }\right)}^{†}{A}_1^{\varphi },C_{22}=A_1{A}_0^{†}{C}_0{\left(A_0^{\varphi }\right)}^{†}{A}_0^{\varphi },\hfill \\ \hfill & C_2=C_1-C_{11}-C_{22},M=R_{A_0}{B}_2^{\varphi },N=R_{A_0}{C}_2.\hfill \end{array}$$

Then the following statements are equivalent:

$\left(1\right)$ The dual quaternion matrix equation (3) is consistent.

$\left(2\right)$ The following equalities are satisfied:

$$\begin{array}{c}\hfill R_{A_0}{C}_0=0,R_{M}N=0,R_{A_0}{C}_2{L}_{A_0^{\varphi }}=0.\end{array}$$

$\left(3\right)$ The following rank equalities hold:

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cc}{A}_0& C_0\end{array}\right)=r\left(\begin{array}{c}{A}_0\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccc}{A}_1& A_0& C_1\\ A_0& 0& C_0\end{array}\right)=r\left(\begin{array}{cc}{A}_1& A_0\\ A_0& 0\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{cc}{C}_1& A_0\\ A_0^{\varphi }& 0\end{array}\right)=r\left(\begin{array}{c}{A}_0\end{array}\right)+r\left(\begin{array}{c}{A}_0^{\varphi }\end{array}\right)=2r\left(A_0\right).\hfill \end{array}$$

In this case, the general solution X of (3) can be expressed as $X=X_0+X_1ϵ$, where

$$X_0=\frac{\widetilde{X_0}+{\widetilde{X_0}}^{\varphi }}{2},X_1=\frac{\widetilde{X_1}+{\widetilde{X_1}}^{\varphi }}{2}$$

and

$$\begin{array}{cc}\hfill & X_0=A_0^{†}{C}_0{\left(A_0^{\varphi }\right)}^{†}+L_{A_0}U+V{R}_{A_0^{\varphi }},\hfill \\ \hfill & X_1=A_0^{†}(C_2-A_{0}V{B}_2-B_2^{\varphi }U{A}_0^{\varphi }){\left(A_0^{\varphi }\right)}^{†}+W_1{R}_{A_0^{\varphi }}+L_{A_0}{W}_2,\hfill \\ \hfill & U=M^{†}N{\left(A_0^{\varphi }\right)}^{†}+L_M{Q}_1+Q_2{R}_{A_0^{\varphi }},\hfill \\ \hfill & V=A_0^{†}{N}^{\varphi }{\left(M^{\varphi }\right)}^{†}+L_{A_0}{Q}_3+Q_4{R}_{M^{\varphi }},\hfill \end{array}$$

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

Proof. 

By using the definitions of equality of dual quaternion matrices and dual quaternion matrix multiplication, we can conclude that the consistency of the dual quaternion matrix equation (3) is contingent on the existence of the solutions to the system of quaternion matrix equations

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & A_0\widetilde{X_0}{A}_0^{\varphi }=C_0,\hfill \\ \hfill & A_0\widetilde{X_0}{A}_1^{\varphi }+A_0\widetilde{X_1}{A}_0^{\varphi }+A_1\widetilde{X_0}{A}_0^{\varphi }=C_1.\hfill \end{array}\right.\end{array}$$

(33)

In fact, if the matrix equation (3) has a $\varphi$-Hermitian solution $X=X_0+X_1ϵ$, it is obvious that $X_0$ and $X_1$ must be solutions to (33). Conversely, if the system (33) has solutions $\widetilde{X_0}$ and $\widetilde{X_1}$, then the matrix equation (3) has solution $X=X_0+X_1ϵ$, where

$$X_0=\frac{\widetilde{X_0}+{\widetilde{X_0}}^{\varphi }}{2},X_1=\frac{\widetilde{X_1}+{\widetilde{X_1}}^{\varphi }}{2}.$$

According to Theorem 1, we can present the necessary and sufficient conditions for the solvability of (33), along with the general expression for its solutions. □

5. Numerical example

Now, we give a numerical example to illustrate the main results of this paper.

Example 1. 

Given the dual quaternion matrices:

$$\begin{array}{cc}\hfill A& =A_0+A_1ϵ=\left(\begin{array}{cc}2\mathbf{i}+\mathbf{k}& 0\\ \mathbf{j}& 0\end{array}\right)+\left(\begin{array}{cc}2-3\mathbf{i}+\mathbf{k}& \mathbf{i}-\mathbf{j}\\ -3\mathbf{k}& \mathbf{i}\end{array}\right)ϵ,\hfill \\ \hfill B& =B_0+B_1ϵ=\left(\begin{array}{cc}1+\mathbf{i}+\mathbf{j}& -\mathbf{j}\\ 0& \mathbf{k}+\mathbf{j}\end{array}\right)+\left(\begin{array}{cc}-\mathbf{i}-2\mathbf{k}& \mathbf{j}+3\mathbf{k}\\ \mathbf{i}& \mathbf{k}\end{array}\right)ϵ,\hfill \\ \hfill C& =C_0+C_1ϵ\hfill \\ \hfill & =\left(\begin{array}{cc}-2+\mathbf{i}+\mathbf{j}+3\mathbf{k}& -1+2\mathbf{i}-2\mathbf{j}-4\mathbf{k}\\ -1+\mathbf{j}-\mathbf{k}& 2+\mathbf{i}\end{array}\right)+\left(\begin{array}{cc}11-5\mathbf{i}+5\mathbf{j}-3\mathbf{k}& -3-8\mathbf{j}+8\mathbf{k}\\ 1+\mathbf{i}-5\mathbf{j}-\mathbf{k}& 2-3\mathbf{i}-\mathbf{j}+\mathbf{k}\end{array}\right)ϵ.\hfill \end{array}$$

Computing directly yields

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cc}{A}_0& C_0\end{array}\right)=r\left(A_0\right)=1,r\left(\begin{array}{c}{B}_0\\ C_0\end{array}\right)=r\left(B_0\right)=2,\hfill \\ \hfill & r\left(\begin{array}{ccc}{A}_1& A_0& C_1\\ A_0& 0& C_0\end{array}\right)=r\left(\begin{array}{cc}{A}_1& A_0\\ A_0& 0\end{array}\right)=3,\hfill \\ \hfill & r\left(\begin{array}{cc}{C}_1& A_0\\ B_0& 0\end{array}\right)=r\left(\begin{array}{c}{A}_0\end{array}\right)+r\left(\begin{array}{c}{B}_0\end{array}\right)=3,\hfill \\ \hfill & r\left(\begin{array}{cc}{B}_1& B_0\\ B_0& 0\\ C_1& C_0\end{array}\right)=r\left(\begin{array}{cc}{B}_1& B_0\\ B_0& 0\end{array}\right)=4.\hfill \end{array}$$

All rank equations are satisfied and the general solution of dual quaternion matrix equation (1) can be expressed as

$$\begin{array}{c}\hfill X=X_0+X_1ϵ=\left(\begin{array}{cc}1& \mathbf{i}\\ \mathbf{k}& 0\end{array}\right)+\left(\begin{array}{cc}0& \mathbf{k}\\ \mathbf{i}& 0\end{array}\right)ϵ.\end{array}$$

6. Conclusions

In this paper, we have established the solvability conditions for the dual quaternion matrix equation (1) by using Moore-Penrose inverses and ranks of matrices, we also have derived the expressions of its general solution to (1) when the solvability conditions are met. As special cases, some dual quaternion matrix equations are also discussed. Moreover, we have investigated the $\varphi$-Hermitian matrix over dual quaternion algebra and provided its related properties, as an application of the aforementioned research, we consider a special case of (1) and give the $\varphi$-Hermitian solutions to (3). Finally, we have presented an example to illustrate the main results.