A Sylvester-Type Matrix Equation over the Dual Quaternion Algebra with an Application

1. Introduction

In 1843, Hamilton proposed the concept of quaternion [1]. After more than a hundred years of development, quaternions have been widely used in many fields up to now, including color face recognition, aerospace, system control, robotics industry, artificial intelligence [2,3,4,5,6] and so on. The set of quaternions, $\mathbb{R}$ denotes the field of real numbers, denoted by $\mathbb{H}$ as follows,

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

which is a non-commutative division ring. Beyond its theoretical foundations, recent work has demonstrated that quaternions offer a powerful tool for model optimization and solving [7]. Furthermore, advances in quaternion matrix calculations point to its broader applicability in emerging domains [8]. In addition, solving quaternion matrix equations is an important issue in many fields such as scientific computing and engineering applications. For more research on quaternions and quaternion matrix equations, please refer to [38,39,40,41,42,48].

Later, Clifford introduced the concept of dual numbers and dual quaternions [9]. Dual quaternions offer a unified mathematical representation for both the rotation and translation components of 3D rigid body motion, making them indispensable in numerous areas of mathematics and physics [10,11,12,13,14,15,16,17]. Dual quaternion matrix equations have also been investigated by a number of scholars and the scope of the research is extensive [44,45,46,47].

As an important type of matrix equations, the Sylvester and Sylvester-type matrix equations have been applied in many fields such as system design [18], control theory [19], sensitivity analysis [20] and data transmission [43]. For this reason, they have attracted the attention of many scholars. Early fundamental research on matrix equations was carried out in real and complex fields. The Sylvester-type matrix equation

$$A_{1}X+Y{B}_1=C_1,$$

(1)

was studied by Roth [21] over polynomial integer rings, while Baksalary and Kalamany [22] focused on its solvability and the structure of its general solutions, then He [23] made further investigations into matrix Equation (1). As research progresses, the investigations on Sylvester-type matrix equations have been extended to $\mathbb{H}$ and $\mathbb{DH}$ in the last decades [49,52]. Scholars have deeply explored generalized Sylvester equations, and various extended matrix systems, and systematically studied solvability criteria, general solutions and constrained solutions such as Hermitian and positive definite solutions by means of generalized inverse, matrix rank method and iterative algorithm, which further enriched the theoretical connotation of non-commutative matrix equations [24,25,26,27,28,29,30]. For dual quaternion matrix equations. Xie and Wang [31] derived the solvability conditions for matrix Equation (1) , and presented the general form of its solution under solvable conditions. This result is of considerable importance to the study of matrix equations. Furthermore, for dual quaternion matrix equation

$$A_{11}X+B_{11}Y{C}_{11}=D_{11},$$

(2)

Li and Wang [32] investigated both this equation and another symmetric counterpart. Subsequent researches further expanded to bilateral coupled equations and constrained solution problems, but most of the research objects are limited to single-variable or two-variable simple structures [30,32,50,51]. The research on dual quaternion matrix equations with multiple unknown matrices is still relatively scarce, and the relevant theoretical achievements need to be further supplemented and perfected.

Motivated by the studies of the above equations and based on Equations (1) and (2), in this paper, we focus on dual quaternion matrix equation

$$AX+BYC+ZD=E.$$

(3)

We adopt Moore-Penrose inverse and rank equalities for matrices to investigate the solvability conditions for Equation (3) and, when it is solvable, to present its general solution expression.

The structure of this paper is organized as follows: Section 2 is devoted to several key definitions and lemmas. In Section 3, the main theorems concerning the dual quaternion matrix equation (3) are formally established. In Section 4, a numerical example is given to validate the main results. Their application to color image encryption and decryption is discussed in Section 5. Finally, Section 6 offers a concise summary of the paper.

2. Preliminaries

In this section, we introduce some fundamental definitions and properties of quaternions and dual quaternions, and present several lemmas that are useful for solving equations.

In this paper, all $m\times n$ matrices over $\mathbb{H}$ is denoted by ${\mathbb{H}}^{m\times n}$ , $r\left(A\right)$ denotes the rank of A. In general, $A^{†}$ stands for the Moore-Penrose inverse of $A\in {\mathbb{H}}^{l\times k}$ , which is defined as the solution of

$$AYA=A,YAY=Y,{\left(AY\right)}^{*}=AY,{\left(YA\right)}^{*}=YA.$$

Moreover, $L_A=I-A^{†}A$ and $R_A=I-A{A}^{†}$ are two projectors associated with A . I and 0 denote the identity matrix and zero matrix with appropriate sizes.

Definition 2.1([33]): Let $x_0,x_1\in \mathbb{H}$. We say x is a dual quaternion if x has the form

$$x=x_0+x_1ϵ,$$

where $ϵ$ is the infinitesimal unit, satisfying ${ϵ}^2=0$, $x_0,x_1$ as the real part and the dual part of x , respectively.

The set of dual quaternions is denoted by

$$\mathbb{DH}=\{x=x_0+x_1ϵ\mid x_0,x_1\in \mathbb{H}\}.$$

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

$${\mathbb{DH}}^{m\times n}=\{X=X_0+X_1ϵ\mid X_0,X_1\in {\mathbb{H}}^{m\times n}\}.$$

For $Y=Y_0+Y_1ϵ\in {\mathbb{DH}}^{m\times n}$, we have $X=Y$ if $X_0=Y_0$ and $X_1=Y_1$ ; furthermore,

$$X+Y=X_0+Y_0+(X_1+Y_1)ϵ,XY=X_0{Y}_0+(X_0{Y}_1+X_1{Y}_0)ϵ.$$

Lemma 2.1([34]): Let A, B, and C be arbitrary matrices over $\mathbb{H}$ with appropriate dimensions. Then, the following equalities hold:

(i)

$R_{A}A=0,A{L}_A=0,A^{†}{R}_A=0,L_A{A}^{†}=0$.

(ii)

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

(iii)

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

Lemma 2.2([35]): Suppose that $A\in {\mathbb{H}}^{m\times s}$, $B\in {\mathbb{H}}^{s\times n}$, and $C\in {\mathbb{H}}^{m\times n}$. Then, the quaternion matrix equation $AXB=C$ is consistent if and only if

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

In this case, the general solution can be expressed as

$$X=A^{†}C{B}^{†}+L_A{U}_1+U_2{R}_B,$$

where $U_1,U_2\in {\mathbb{H}}^{s\times t}$ are arbitrary.

Lemma 2.3 ([22]): Let $A_1,B_1$ and $C_1$ be given matrices with suitable sizes. Then, the Sylvester-type Equation $A_{1}X+Y{B}_1=C_1$ is solvable if and only if

$$R_{A_1}{C}_1{L}_{B_1}=0.$$

In this case, the general solution to the Equation can be expressed as

$$X=A_1^{†}{C}_1-A_1^{†}{U}_1{B}_1+L_{A_1}{U}_2,Y=R_{A_1}{C}_1{B}_1^{†}+A_1{A}_1^{†}{U}_1+U_3{R}_{B_1},$$

where $U_1,U_2,$ and $U_3$ are arbitrary matrices with appropriate sizes.

For clarity in block matrix notation, we use commas as separators. For example, the block matrix $\left(AB\right)$ is denoted as $(A,B)$ .

Lemma 2.4 ([36]): Let $A_1,A_2\in {\mathbb{H}}^{m\times s}$, $B_1,B_2\in {\mathbb{H}}^{s\times n}$, and $C\in {\mathbb{H}}^{m\times n}$ . Set

$$E=R_{A_1}{A}_2,F=R_{A_1}C,M=B_1{L}_{B_2},N=C{L}_{B_2}.$$

Then, the following descriptions are equivalent.

(i)

The quaternion matrix equation

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

is consistent.

(ii)

$R_{E}F=0$, $R_{A_1}C{L}_{B_2}=0$, $N{L}_M=0$ .

(iii)

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

In this case, the general solution to the Equation can be expressed as

$$\begin{array}{cc}\hfill Y_1& =A_1^{†}N{M}^{†}+L_{A_1}{W}_1+W_4{R}_M,\hfill \\ \hfill Y_2& =A_1^{†}\left(C-A_1{Y}_1{B}_1-A_2{Y}_3{B}_2\right)B_2^{†}+L_{A_1}{W}_5+W_6{R}_{B_2},\hfill \\ \hfill Y_3& =E^{†}F{B}_2^{†}+L_E{W}_1+W_2{R}_{B_2},\hfill \end{array}$$

where $W_i(i=\overline{1,6})$ are arbitrary matrices over $\mathbb{H}$ with appropriate sizes.

Lemma 2.5 ([37]): Assume that $A\in {\mathbb{H}}^{m\times n}$, $B\in {\mathbb{H}}^{n\times s}$, $C\in {\mathbb{H}}^{s\times n}$, $D\in {\mathbb{H}}^{s\times n}$, and $E\in {\mathbb{H}}^{s\times p}$ . Then, the following statements hold.

(i)

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

3. General Solutions of Equation (3)

In this section, we first establish the necessary and sufficient conditions for the solvability of the dual quaternion matrix equation (3). We then provide their general solution expressions under the condition of solvability.

Theorem 3.1: Let $A=A_0+A_1ϵ\in \mathbb{D}{\mathbb{H}}^{m\times t}$, $B=B_0+B_1ϵ\in \mathbb{D}{\mathbb{H}}^{m\times p}$, $C=C_0+C_1ϵ\in \mathbb{D}{\mathbb{H}}^{s\times n}$, $D=D_0+D_1ϵ\in \mathbb{D}{\mathbb{H}}^{q\times n}$, $E=E_0+E_1ϵ\in \mathbb{D}{\mathbb{H}}^{m\times n}$. Denote

$$\begin{array}{cc}\hfill T_1& =R_{A_0}{B}_0,T_2=C_0{L}_{D_0},T_3=R_{A_0}{A}_1{A}_0^{†}{B}_0-R_{A_0}{B}_1,T_4=R_{A_0}{T}_{10}{L}_{D_0},T=-R_{T_6}{T}_4{L}_{T_9},\hfill \\ \hfill T_5& =R_{T_2}{C}_0{D}_0^{†}{D}_1{L}_{D_0}-R_{T_2}{C}_1{L}_{D_0},T_6=R_{A_0}{A}_1{L}_{A_0},T_7=L_{D_0},T_8=R_{T_6}{R}_{A_0},\hfill \\ \hfill A_{11}& =R_{T_6}{T}_1,B_{11}=T_5{L}_{T_9},A_{22}=R_{T_6}{T}_3{L}_{T_1},B_{22}=T_2{L}_{T_9},T_9=R_{D_0}{D}_1{L}_{D_0},M=R_{A_{11}}{A}_{22},\hfill \\ \hfill T_{10}& =E_1-A_1{A}_0^{†}{E}_0+\left(A_1{A}_0^{†}{B}_0-B_1\right)T_1^{†}{E}_0{T}_2^{†}{C}_0-B_0{T}_1^{†}{E}_0{T}_2^{†}{C}_1-R_{A_0}{E}_0{D}_0^{†}{D}_1+T_1{T}_1^{†}{E}_0{T}_2^{†}{C}_0{D}_0^{†}{D}_1,\hfill \\ \hfill T_{11}& =-T_{10}+A_0{A}_0^{†}(E_0{D}_0^{†}{D}_1-B_0{T}_1^{†}{E}_0{T}_2^{†}{C}_0{D}_0^{†}{D}_1),N=R_{A_{11}}T,P=B_{11}{L}_{B_{22}},Q=T{L}_{B_{22}}.\hfill \end{array}$$

Then, the following statements are equivalent.

(i) The dual quaternion matrix equation (3) is consistent.

(ii)

$$\begin{array}{c}\hfill R_{T_1}{R}_{A_0}{E}_0{L}_{D_0}=0,R_{A_0}{E}_0{L}_{D_0}{L}_{T_2}=0,\end{array}$$

(4)

$$\begin{array}{c}\hfill R_{M}N=0,R_{A_{11}}T{L}_{B_{22}}=0,Q{L}_P=0.\end{array}$$

(5)

(iii)

$$\begin{array}{c}\hfill r\left(\begin{array}{ccc}{A}_0& B_0& D_0\\ 0& 0& E_0\end{array}\right)=r(A_0,B_0)+r\left(D_0\right),r\left(\begin{array}{cc}{E}_0& A_0\\ C_0& 0\\ D_0& 0\end{array}\right)=r\left(\begin{array}{c}{C}_0\\ D_0\end{array}\right)+r\left(A_0\right),\end{array}$$

(6)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccccccc}{A}_0& A_1& B_0& 0& -B_1& -E_0& -E_1\\ 0& A_0& 0& 0& 0& 0& -E_0\\ 0& 0& 0& A_0& B_0& 0& E_0\\ 0& 0& 0& 0& 0& D_0& D_1\\ 0& 0& 0& 0& 0& 0& D_0\end{array}\right)=r\left(\begin{array}{ccccc}{A}_1& A_1& B_0& 0& -B_1\\ 0& A_0& 0& 0& 0\\ 0& 0& 0& A_0& B_0\end{array}\right)+r\left(\begin{array}{cc}{D}_1& D_0\\ D_0& 0\end{array}\right)\hfill \\ \hfill & r\left(\begin{array}{ccccc}{A}_0& A_1& -E_0& -E_1& B_0\\ 0& A_0& 0& -E_0& 0\\ 0& 0& 0& C_0& 0\\ 0& 0& D_0& D_1& 0\\ 0& 0& 0& D_0& 0\end{array}\right)=r\left(\begin{array}{ccc}{A}_0& A_1& B_0\\ 0& A_0& 0\end{array}\right)+r\left(\begin{array}{cc}{C}_0& 0\\ D_1& D_0\\ D_0& 0\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccccc}{A}_0& A_1& E_0& -E_0& -E_1\\ 0& A_0& 0& 0& -E_0\\ 0& 0& C_0& 0& -C_1\\ 0& 0& D_0& 0& 0\\ 0& 0& 0& 0& C_0\\ 0& 0& 0& D_0& D_1\\ 0& 0& 0& 0& D_0\end{array}\right)=r\left(\begin{array}{ccc}{C}_0& 0& -C_1\\ D_0& 0& 0\\ 0& 0& C_0\\ 0& D_0& D_1\\ 0& 0& D_0\end{array}\right)+r\left(\begin{array}{cc}{A}_1& A_0\\ A_0& 0\end{array}\right).\hfill \end{array}$$

(7)

In this case, the general solution X, Y and Z of equation (3) can be expressed as $X=X_0+X_1ϵ$, $Y=Y_0+Y_1ϵ$, $Z=Z_0+Z_1ϵ$, where

$$\begin{array}{cc}\hfill X_0& =A_0^{†}(E_0-B_0{Y}_0{C}_0)-A_0^{†}{V}_0{D}_0+L_{A_0}{W}_1,\hfill \\ \hfill Y_0& =T_1^{†}{E}_0{T}_2^{†}+L_{T_1}{W}_3+W_4{R}_{T_2},\hfill \\ \hfill Z_0& =R_{A_0}(E_0-B_0{Y}_0{C}_0)D_0^{†}+A_0{A}_0^{†}{V}_0+W_2{R}_{D_0},\hfill \\ \hfill X_1& =A_0^{†}(T_{10}+U)-A_0^{†}{V}_{11}{D}_0+L_{A_0}{V}_{12},\hfill \\ \hfill Y_1& =A_{11}^{†}(A_{11}{W}_4{B}_{11}+A_{22}{W}_3{B}_{22}-T)B_{22}^{†}+L_{A_{11}}{V}_5+V_6{R}_{B_{22}},\hfill \\ \hfill Z_1& =R_{A_0}(T_{10}+U)D_0^{†}+A_0{A}_0^{†}{V}_{11}+V_{13}{R}_{D_0}\hfill \\ \hfill W_1& =T_6^{†}(T_4+T_3{L}_{T_1}{W}_3{T}_2-T_1{Y}_1{T}_2+T_1{W}_4{T}_5-R_{A_0}{W}_2{R}_{D_0}{D}_1{L}_{D_0})T_7^{†}+L_{T_6}{V}_9+V_{10}{R}_{T_7},\hfill \\ \hfill W_2& =T_8^{†}(R_{T_6}{T}_4+R_{T_6}{T}_3{L}_{T_1}{W}_3{T}_2-R_{T_6}{T}_1{Y}_1{T}_2+R_{T_6}{T}_1{W}_4{T}_5)T_9^{†}+L_{T_8}{V}_7+V_8{R}_{T_9},\hfill \\ \hfill W_3& =M^{†}N{B}_{22}^{†}+L_M{V}_1+V_2{R}_{B_{22}},\hfill \\ \hfill W_4& =A_{11}^{†}Q{P}^{†}+L_{A_{11}}{V}_3+V_4{R}_P,\hfill \\ \hfill U& =A_1{A}_0^{†}{V}_0{D}_0-A_0{A}_0^{†}{V}_0{D}_1-A_1{L}_{A_0}{W}_1-W_2{R}_{D_0}{D}_1+(A_1{A}_0^{†}{B}_0-B_1)L_{T_1}{W}_3{C}_0-B_0{L}_{T_1}{W}_3{C}_1\hfill \\ \hfill & +(A_1{A}_0^{†}{B}_0-B_1)W_4{R}_{T_2}{C}_0-B_0{W}_4{R}_{T_2}{C}_1+T_1{W}_4{R}_{T_2}{C}_0{D}_0^{†}{D}_1-B_0{Y}_1{C}_0,\hfill \end{array}$$

(8)

with arbitrary $V_i(i=\overline{0,13})$.

Proof. The entire proof process consists of two steps: first, we prove that (i) ⇔ (ii) and derive the expression for the general solution; then, prove that (ii) ⇔ (iii) .

We assume that Equation (3) is solvable with solutions $X\in \mathbb{D}{\mathbb{H}}^{t\times n}$ , $Y\in \mathbb{D}{\mathbb{H}}^{p\times s}$ and $Z\in \mathbb{D}{\mathbb{H}}^{m\times q}$, which can be expressed as

$$X=X_0+X_1ϵ,Y=Y_0+Y_1ϵ,Z=Z_0+Z_1ϵ.$$

Substituting these expressions into the Equation (3), we obtain that the dual quaternion matrix Equation (3) is equivalent to the following system of quaternion matrix equations

$$\left\{\begin{array}{c}{A}_0{X}_0+B_0{Y}_0{C}_0+Z_0{D}_0=E_0,\hfill \\ A_1{X}_0+A_0{X}_1+B_1{Y}_0{C}_0+B_0{Y}_1{C}_0+B_0{Y}_0{C}_1+Z_0{D}_1+Z_1{D}_0=E_1.\hfill \end{array}\right.$$

(9)

Next, we prove that (i) ⇔ (ii), and illustrate the general solution of (9) with the form of (8).

We decompose the system (9) into the following equations

$$A_0{X}_0+B_0{Y}_0{C}_0+Z_0{D}_0=E_0$$

(10)

and

$$A_1{X}_0+A_0{X}_1+B_1{Y}_0{C}_0+B_0{Y}_1{C}_0+B_0{Y}_0{C}_1+Z_0{D}_1+Z_1{D}_0=E_1$$

(11)

By manipulating the Equation (10), we obtain

$$A_0{X}_0+Z_0{D}_0=E_0-B_0{Y}_0{C}_0.$$

(12)

From Lemma 2.3, we conclude that (12) is consistent if and only if the following condition holds

$$R_{A_0}(E_0-B_0{Y}_0{C}_0)L_{D_0}=0,$$

(13)

which is equivalent to

$$R_{T_1}{R}_{A_0}{E}_0{L}_{D_0}=0,R_{A_0}{E}_0{L}_{D_0}{L}_{T_2}=0$$

In this case, the general solution to (10) can be written as

$$\begin{array}{cc}\hfill X_0& =A_0^{†}(E_0-B_0{Y}_0{C}_0)-A_0^{†}{V}_0{D}_0+L_{A_0}{W}_1,\hfill \\ \hfill Y_0& =T_1^{†}{E}_0{T}_2^{†}+L_{T_1}{W}_3+W_4{R}_{T_2},\hfill \\ \hfill Z_0& =R_{A_0}(E_0-B_0{Y}_0{C}_0)D_0^{†}+A_0{A}_0^{†}{V}_0+W_2{R}_{D_0}.\hfill \end{array}$$

(14)

Substituting (14) into (11) and simplifying, we obtain

$$A_0{X}_1+Z_1{D}_0=T_{10}+U.$$

(15)

By applying Lemma 2.3, we conclude that (15) is consistent if and only if

$$R_{A_0}(T_{10}+U)L_{D_0}=0.$$

(16)

Simplifying (16), we obtain

$$R_{A_0}{A}_1{L}_{D_0}{W}_1{L}_{D_0}=T_4+T_3{L}_{T_1}{W}_3{T}_2-T_1{Y}_1{T}_2+T_1{W}_4{T}_5-R_{A_0}{W}_2{R}_{D_0}{D}_1{L}_{D_0}.$$

(17)

(17) is consistent if and only if

$$R_{T_6}{R}_{A_0}{W}_2{R}_{D_0}{D}_1{L}_{D_0}=R_{T_6}{T}_4+R_{T_6}{T}_3{L}_{T_1}{W}_3{T}_2-R_{T_6}{T}_1{Y}_1{T}_2+R_{T_6}{T}_1{W}_4{T}_5.$$

(18)

This leads to the consistency condition

$$A_{11}{W}_4{B}_{11}-A_{11}{Y}_1{B}_{22}+A_{22}{W}_3{B}_{22}=T.$$

(19)

By Lemma 2.4, we conclude that (19) is consistent if and only if

$$R_{M}N=0,R_{A_{11}}T{L}_{B_{22}}=0,Q{L}_P=0.$$

In this case, the general solution of (19) can be written as

$$\begin{array}{cc}\hfill W_3& =M^{†}N{B}_{22}^{†}+L_M{V}_1+V_2{R}_{B_{22}},\hfill \\ \hfill W_4& =A_{11}^{†}Q{P}^{†}+L_{A_{11}}{V}_3+V_4{R}_P,\hfill \\ \hfill Y_1& =A_{11}^{†}(A_{11}{W}_4{B}_{11}+A_{22}{W}_3{B}_{22}-T)B_{22}^{†}+L_{A_{11}}{V}_5+V_6{R}_{B_{22}}\hfill \end{array}$$

(20)

Substituting (20) into (18), and applying Lemma 2.2, the general solution for $W_2$ is

$$W_2=T_8^{†}(R_{T_6}{T}_4+R_{T_6}{T}_3{L}_{T_1}{W}_3{T}_2-R_{T_6}{T}_1{Y}_1{T}_2+R_{T_6}{T}_1{B}_0{W}_4{T}_5)T_9^{†}+L_{T_8}{V}_7+V_8{R}_{T_9}$$

(21)

Substituting (20) and (21) into (17), and using Lemma 2.2, the general solution for $W_1$ can be written as

$$W_1=T_6^{†}(T_4+T_3{L}_{T_1}{W}_3{T}_2-T_1{Y}_1{T}_2+T_1{W}_4{T}_5-R_{A_0}{W}_2{R}_{D_0}{D}_1{L}_{D_0})T_7^{†}+L_{T_6}{V}_9+V_{10}{R}_{T_7}$$

(22)

Substituting (20) , (21) and (22) into (15), and again using Lemma 2.3, the general solution for $X_1$ and $Z_1$ can be written as

$$\begin{array}{cc}\hfill X_1& =A_0^{†}(T_{10}+U)-A_0^{†}{V}_{11}{D}_0+L_{A_0}{V}_{12}\hfill \\ \hfill Z_1& =R_{A_0}(T_{10}+U)D_0^{†}+A_0{A}_0^{†}{V}_{11}+V_{13}{R}_{D_0}\hfill \end{array}$$

(23)

where $V_i(i=\overline{0,13})$ are arbitrary matrices over $\mathbb{H}$ with appropriate dimensions.

In summary, we have proven the equivalence of (i) and (ii) . Additionally, we have derived the expression for the general solution.

Finally, we prove the equivalence between (ii) and (iii). According to Lemma 2.5, it can be verified that (4) ⇔ (6).

Now, we turn to prove that (5) ⇔ (7) . We take the particular solution as follows:

$$\begin{array}{c}\hfill X_0=A_0^{†}(E_0-B_0{T}_1^{†}{E}_0{T}_2^{†}{C}_0),Y_0=T_1^{†}{E}_0{T}_2^{†},Z_0=R_{A_0}(E_0-B_0{T}_1^{†}{E}_0{T}_2^{†}{C}_0)D_0^{†}.\end{array}$$

It is straightforward to verify that $X_0$ , $Y_0$ and $Z_0$ form a particular solution to the quaternion matrix equation $A_0{X}_0+B_0{Y}_0{C}_0+Z_0{D}_0=E_0$ . Now, applying Lemma 2.5, we obtain

$$\begin{array}{cc}\hfill R_{M}N=0& \iff r\left(R_{M}N\right)=0\hfill \\ & \iff r(M,N)=r\left(M\right)\hfill \\ & \iff r(A_{11},A_{22},T)=r(A_{11},A_{22})\hfill \\ & \iff r(T_6,T_1,T_3{L}_{T_1},-T_4{L}_{T_9})=r(T_6,T_1,T_3{L}_{T_1})\hfill \\ & \iff r\left(\begin{array}{cccc}{T}_6& T_1& T_3& -T_4\\ 0& 0& T_1& 0\\ 0& 0& 0& T_9\end{array}\right)=r\left(\begin{array}{ccc}{T}_6& T_1& T_3\\ 0& 0& T_1\end{array}\right)+r\left(T_9\right)\hfill \\ & \iff r\left(\begin{array}{ccccccc}{A}_0& A_1& B_0& 0& A_1{A}_0^{†}{B}_0-B_1& 0& T_{11}\\ 0& A_0& 0& 0& 0& 0& 0\\ 0& 0& 0& A_0& B_0& 0& 0\\ 0& 0& 0& 0& 0& D_0& D_1\\ 0& 0& 0& 0& 0& 0& D_0\end{array}\right)-r\left(A_0\right)\hfill \\ & =r\left(\begin{array}{ccccc}{A}_0& A_1{L}_{A_0}& B_0& 0& A_1{A}_0^{†}{B}_0-B_1\\ 0& 0& 0& A_0& B_0\end{array}\right)+r\left(\begin{array}{cc}{D}_1& D_0\\ D_0& 0\end{array}\right)\hfill \\ & \iff r\left(\begin{array}{ccccccc}{A}_0& A_1& B_0& 0& A_1{A}_0^{†}{B}_0-B_1& 0& -E_1+A_1{A}_0^{†}{E}_0+E_0{D}_0^{†}{D}_1\\ 0& A_0& 0& 0& 0& 0& 0\\ 0& 0& 0& A_0& B_0& 0& B_0{T}_1^{†}{E}_0{T}_2^{†}{C}_1\\ 0& 0& 0& 0& 0& D_0& D_1\\ 0& 0& 0& 0& 0& 0& D_0\end{array}\right)\hfill \\ & =r\left(\begin{array}{ccccc}{A}_0& A_1& B_0& 0& A_1{A}_0^{†}{B}_0-B_1\\ 0& A_0& 0& 0& 0\\ 0& 0& 0& A_0& B_0\end{array}\right)+r\left(\begin{array}{cc}{D}_1& D_0\\ D_0& 0\end{array}\right)\hfill \\ & \iff r\left(\begin{array}{ccccccc}{A}_0& A_1& B_0& 0& -B_1& -E_0{D}_0^{†}{D}_0& -E_1\\ 0& A_0& 0& 0& 0& 0& -A_0{A}_0^{†}{E}_0\\ 0& 0& 0& A_0& B_0& 0& E_0\\ 0& 0& 0& 0& 0& D_0& D_1\\ 0& 0& 0& 0& 0& 0& D_0\end{array}\right)\hfill \\ & =r\left(\begin{array}{ccccc}{A}_0& A_1& B_0& 0& -B_1\\ 0& A_0& 0& 0& 0\\ 0& 0& 0& A_0& B_0\end{array}\right)+r\left(\begin{array}{cc}{D}_1& D_0\\ D_0& 0\end{array}\right)\hfill \\ & \iff r\left(\begin{array}{ccccccc}{A}_0& A_1& B_0& 0& -B_1& -E_0& -E_1\\ 0& A_0& 0& 0& 0& 0& -E_0\\ 0& 0& 0& A_0& B_0& 0& E_0\\ 0& 0& 0& 0& 0& D_0& D_1\\ 0& 0& 0& 0& 0& 0& D_0\end{array}\right)\hfill \\ & =r\left(\begin{array}{ccccc}{A}_0& A_1& B_0& 0& -B_1\\ 0& A_0& 0& 0& 0\\ 0& 0& 0& A_0& B_0\end{array}\right)+r\left(\begin{array}{cc}{D}_1& D_0\\ D_0& 0\end{array}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill R_{A_{11}}T{L}_{B_{22}}=0& \iff r\left(R_{A_{11}}T{L}_{B_{22}}\right)=0\hfill \\ \hfill & \iff r\left(\begin{array}{cc}T& A_{11}\\ B_{22}& 0\end{array}\right)=r\left(A_{11}\right)+r\left(B_{22}\right)\hfill \\ \hfill & \iff r\left(\begin{array}{ccc}{T}_6& -T_4& T_1\\ 0& T_2& 0\\ 0& T_9& 0\end{array}\right)=r(T_6,T_1)+r\left(\begin{array}{c}{T}_2\\ T_9\end{array}\right)\hfill \\ \hfill & \iff r\left(\begin{array}{ccc}{A}_1{L}_{A_0}& T_{11}{L}_{D_0}& B_0\\ 0& C_0{L}_{D_0}& 0\\ 0& R_{D_0}{D}_1{L}_{D_0}& 0\end{array}\right)=r\left(\begin{array}{ccc}{A}_0,& A_1{L}_{A_0},& B_0\end{array}\right)+r\left(\begin{array}{c}{C}_0{L}_{D_0}\\ R_{D_0}{D}_1{L}_{D_0}\end{array}\right)\hfill \\ \hfill & \iff r\left(\begin{array}{ccccc}{A}_0& A_1& 0& T_{11}& B_0\\ 0& A_0& 0& 0& 0\\ 0& 0& 0& C_0& 0\\ 0& 0& D_0& D_1& 0\\ 0& 0& 0& D_0& 0\end{array}\right)=r\left(\begin{array}{ccc}{A}_0& A_1& B_0\\ 0& A_0& 0\end{array}\right)+r\left(\begin{array}{cc}{C}_0& 0\\ D_1& D_0\\ D_0& 0\end{array}\right)\hfill \\ \hfill & \iff r\left(\begin{array}{ccccc}{A}_0& A_1& -E_0{D}_0^{†}{D}_0& -E_1& B_0\\ 0& A_0& 0& -A_0{A}_0^{†}{E}_0& 0\\ 0& 0& 0& C_0& 0\\ 0& 0& D_0& D_1& 0\\ 0& 0& 0& D_0& 0\end{array}\right)=r\left(\begin{array}{ccc}{A}_0& A_1& B_0\\ 0& A_0& 0\end{array}\right)+r\left(\begin{array}{cc}{C}_0& 0\\ D_1& D_0\\ D_0& 0\end{array}\right)\hfill \\ \hfill & \iff r\left(\begin{array}{ccccc}{A}_0& A_1& -E_0& -E_1& B_0\\ 0& A_0& 0& -E_0& 0\\ 0& 0& 0& C_0& 0\\ 0& 0& D_0& D_1& 0\\ 0& 0& 0& D_0& 0\end{array}\right)=r\left(\begin{array}{ccc}{A}_0& A_1& B_0\\ 0& A_0& 0\end{array}\right)+r\left(\begin{array}{cc}{C}_0& 0\\ D_1& D_0\\ D_0& 0\end{array}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill Q{L}_P=0& \iff r\left(Q{L}_P\right)=0\hfill \\ \hfill & \iff r\left(\begin{array}{c}Q\\ P\end{array}\right)=r\left(P\right)\hfill \\ \hfill & \iff r\left(\begin{array}{c}T\\ B_{11}\\ B_{22}\end{array}\right)=r\left(\begin{array}{c}{B}_{11}\\ B_{22}\end{array}\right)\hfill \\ \hfill & \iff r\left(\begin{array}{cc}{R}_{A_0}{A}_1{L}_{A_0}& R_{A_0}{T}_{11}{L}_{D_0}\\ 0& R_{T_2}(C_0{D}_0^{†}{D}_1-C_1)L_{D_0}\\ 0& C_0{L}_{D_0}\\ 0& R_{D_0}{D}_1{L}_{D_0}\end{array}\right)=r\left(\begin{array}{c}{R}_{T_2}(C_0{D}_0^{†}{D}_1-C_1)L_{D_0}\\ C_0{L}_{D_0}\\ R_{D_0}{D}_1{L}_{D_0}\end{array}\right)+r\left(T_6\right)\hfill \\ \hfill & \iff r\left(\begin{array}{ccccc}{A}_0& A_1& 0& 0& T_{11}\\ 0& A_0& 0& 0& 0\\ 0& 0& C_0& 0& C_0{D}_0^{†}{D}_1-C_1\\ 0& 0& D_0& 0& 0\\ 0& 0& 0& 0& C_0\\ 0& 0& 0& D_0& D_1\\ 0& 0& 0& 0& D_0\end{array}\right)=r\left(\begin{array}{ccc}{C}_0& 0& -C_1\\ D_0& 0& 0\\ 0& 0& C_0\\ 0& D_0& D_1\\ 0& 0& D_0\end{array}\right)+r\left(\begin{array}{cc}{A}_1& A_0\\ A_0& 0\end{array}\right)\hfill \\ \hfill & \iff r\left(\begin{array}{ccccc}{A}_0& A_1& B_0{T}_1^{†}{E}_0{T}_2^{†}{C}_1& -E_0{D}_0^{†}{D}_0& -E_1\\ 0& A_0& 0& 0& -A_0{A}_0^{†}{E}_0\\ 0& 0& C_0& 0& C_0{D}_0^{†}{D}_1-C_1\\ 0& 0& D_0& 0& 0\\ 0& 0& 0& 0& C_0\\ 0& 0& 0& D_0& D_1\\ 0& 0& 0& 0& D_0\end{array}\right)\hfill \\ \hfill & =r\left(\begin{array}{ccc}{C}_0& 0& -C_1\\ D_0& 0& 0\\ 0& 0& C_0\\ 0& D_0& D_1\\ 0& 0& D_0\end{array}\right)+r\left(\begin{array}{cc}{A}_1& A_0\\ A_0& 0\end{array}\right)\hfill \\ \hfill & \iff r\left(\begin{array}{ccccc}{A}_0& A_1& E_0& -E_0& -E_1\\ 0& A_0& 0& 0& -E_0\\ 0& 0& C_0& 0& -C_1\\ 0& 0& D_0& 0& 0\\ 0& 0& 0& 0& C_0\\ 0& 0& 0& D_0& D_1\\ 0& 0& 0& 0& D_0\end{array}\right)=r\left(\begin{array}{ccc}{C}_0& 0& -C_1\\ D_0& 0& 0\\ 0& 0& C_0\\ 0& D_0& D_1\\ 0& 0& D_0\end{array}\right)+r\left(\begin{array}{cc}{A}_1& A_0\\ A_0& 0\end{array}\right)\hfill \\ \hfill \square \end{array}$$

4. Numerical Example

Section 3 introduces Theorem 3.1 and presents the general solutions for Equation (3) when it is solvable. In this section, based on Theorem 3.1, we provide Algorithm 1 for solving Equation (3) and perform extensive numerical experiments to validate the error.

Algorithm 1 Solving the Equation (3) over dual quaternions

1. Input: The coefficient matrices A, B, C, D andE.

2. By directly calculating the formulas (4) and (5) in Theorem 3.1. If all formulas hold, proceed

to the next step. If any formula does not hold, output ’The equation has no solution’.

3. By using the coefficient matrices A, B, C , D andE determined above, and combining them

with formulas (8) in Theorem 3.1, the general solutions of $X_0$, $X_1$, $Y_0$, and $Y_1$, $Z_0$, and $Z_1$

can be calculated.

4. Output: $X_0$, $X_1$, $Y_0$, and $Y_1$, $Z_0$, $Z_1$ .

For a matrix X, the expression $X(:,i:j)$ denotes the submatrix consisting of all rows of X and the columns from i to j. Next, we present Example to validate the effectiveness of Algorithm 1.

Example: Given dual quaternion matrices $A=A_0+A_1ϵ$, $B=B_0+B_1ϵ$, $C=C_0+C_1ϵ$ and $D=D_0+D_1ϵ$ ,$E=E_0+E_1ϵ\in \mathbb{D}{\mathbb{H}}^{4\times 4}$ as

$$\begin{array}{cc}\hfill A& =A_0+A_1ϵ\hfill \\ & =\left(\begin{array}{cc}0.9-2i-0.8j+0.7k& 0.2+2.8i+0.1j-1.5k\\ 0.2-1.3i+0.3j+1.7k& -0.1-1.4i+0.5j+0.6k\\ -0.6+0.3i-0.1j+0.5k& -0.9-0.2i+1.1j-0.2k\\ -0.8-0.2i-0.5j+0.9k& -0.8-0.9j+0.6k\end{array}\right.\hfill \\ & \left.\begin{array}{cc}0.9-2i-0.8j+0.7k& 0.2+2.8i+0.1j-1.5k\\ 0.2-1.3i+0.3j+1.7k& -0.1-1.4i+0.5j+0.6k\\ -0.6+0.3i-0.1j+0.5k& -0.9-0.2i+1.1j-0.2k\\ -0.8-0.2i-0.5j+0.9k& -0.8-0.9j+0.6k\end{array}\right)\hfill \\ & +\left(\begin{array}{cc}-0.9-0.3i-0.4j+0.6k& 0.3-1.7i-0.9j-2.3k\\ 1.4+0.8i-1.4j-0.2k& -1.7-0.8i+1.4j-0.2k\\ 0.4-0.1i+0.7j-0.2k& 0.8+0.1i+0.8j-1.2k\\ -1.9-0.2i-1.2j-0.8k& 1.4-0.4i-0.8j-0.2k\end{array}\right.\hfill \\ & \left.\begin{array}{cc}-0.9-0.3i-0.4j+0.6k& 0.3-1.7i-0.9j-2.3k\\ 1.4+0.8i-1.4j-0.2k& -1.7-0.8i+1.4j-0.2k\\ 0.4-0.1i+0.7j-0.2k& 0.8+0.1i+0.8j-1.2k\\ -1.9-0.2i-1.2j-0.8k& 1.4-0.4i-0.8j-0.2k\end{array}\right)ϵ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill B& =B_0+B_1ϵ\hfill \\ & =\left(\begin{array}{cc}2.5+1.2i-0.3j+2.3k& -0.7+0.1i+0.8j+1.3k\\ 0.3i+1.3j-0.3k& -1+0.1i-0.1j+0.1k\\ 1.5+0.3i-1.2j+1.4k& -0.6-0.6i-0.4j+1k\\ -1.1-0.8i-0.2j-0.5k& -1-0.2i+0.7j+0.2k\end{array}\right.\hfill \\ & \left.\begin{array}{cc}0.6-1.9i-2.7j-1.7k& -0.7+0.1i+0.8j+1.3k\\ -0.1+0.9i-0.5j-2.4k& -1+0.1i-0.1j+0.1k\\ -1.1-i+1.6j-0.2k& -0.6-0.6i-0.4j+1k\\ -1.2+0.5j+0.3k& -1-0.2i+0.7j+0.2k\end{array}\right)\hfill \\ & +\left(\begin{array}{cc}0.7-0.7i+0.6j+0.4k& -0.3-0.2i+2.3j-2.3k\\ -1+0.5i+2j+0.6k& 0.4+1.1i+0.9j-1.8k\\ -0.1-1.5i-0.4j-0.5k& 1.8+0.4i-0.2j\\ -0.6+1.6i+0.6j-0.7k& 0.9+1.1i-2j+0.3k\end{array}\right.\hfill \\ & \left.\begin{array}{cc}-0.3-0.2i+2.3j-2.3k& -0.7+0.3i+0.2j-0.9k\\ 0.4+1.1i+0.9j-1.8k& -1.4-0.2i+0.3j+0.5k\\ 1.8+0.4i-0.2j& 0.8+1.7i-0.8j+1.3k\\ 0.9+1.1i-2j+0.3k& -1-0.8i+j+0.7k\end{array}\right)ϵ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill C& =C_0+C_1ϵ\hfill \\ & =\left(\begin{array}{cc}-0.2+0.2i-0.5j-0.9k& -0.7-1.8i-0.6j-0.5k\\ 0.3+0.2i+0.6j+1.3k& -0.8-0.1i+0.2j-0.3k\\ -1.7-1.4i+0.5j-0.2k& 1.9-1.6i-0.3j+0.2k\\ -1.1+0.4i+1.1j+0.9k& -0.3-0.2i-0.6j+2k\end{array}\right.\hfill \\ & \left.\begin{array}{cc}-0.2+0.2i-0.5j-0.9k& 0.5-1.2i+0.4j+0.1k\\ 0.3+0.2i+0.6j+1.3k& 0.4+0.3i-0.1j-0.5k\\ -1.7-1.4i+0.5j-0.2k& -1.2+0.6i-1.1j-0.3k\\ -1.1+0.4i+1.1j+0.9k& 1.1+1.6i+0.3j+k\end{array}\right)\hfill \\ & +\left(\begin{array}{cc}1.3+0.3i+0.3j-0.2k& -1.8+1.1i+1.1j-0.9k\\ 1+0.8i-0.5j-1.6k& -0.5+0.2i-0.5j-1.6k\\ -0.6-0.6i-0.8j-0.7k& 0.3+1.3i+0.1j-0.7k\\ -1.3+0.5i-2.6j+0.8k& -1-0.9i-1.3j-0.9k\end{array}\right.\hfill \\ & \left.\begin{array}{cc}2.1+0.3i+0.1j+0.7k& -1.8+1.1i+1.1j-0.9k\\ -0.2-0.9i-1.2j-2.4k& -0.5+0.2i-0.5j-1.6k\\ -0.6-0.7i-0.9j-1.8k& 0.3+1.3i+0.1j-0.7k\\ -1+0.4i-0.8j-0.8k& -1-0.9i-1.3j-0.9k\end{array}\right)ϵ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill D& =D_0+D_1ϵ\hfill \\ & =\left(\begin{array}{cc}0.6+0.2i+0.2j+0.5k& -0.7+1.2i-3.7j+0.8k\\ -0.4-1.7i+0.2j+0.2k& 0.3+0.3i-1.7j-1.9k\\ -0.6-0.7j-0.3k& -0.2+0.1i-0.8j+0.4k\\ -0.9+1.2i+1.2j& -1.5+1.4i-1.3j-0.5k\end{array}\right.\hfill \\ & \left.\begin{array}{cc}0.6+0.2i+0.2j+0.5k& 0.8+1.2i-0.8j+0.6k\\ -0.4-1.7i+0.2j+0.2k& -0.1+0.3i-0.3j+0.7k\\ -0.6-0.7j-0.3k& -1.6-1.7i-0.7j-0.7k\\ -0.9+1.2i+1.2j& 2.2+1.1i-0.9j-1.3k\end{array}\right)\hfill \\ & +\left(\begin{array}{cc}-0.4-0.7i+0.4j-1.5k& -1-0.6i-0.5j-0.8k\\ 1+0.1i+0.9j+1.3k& 0.5-0.3i+0.5j+0.1k\\ 0.3+i+0.2j-0.7k& 1.4+1.2i-0.3j+1.2k\\ 0.4+1.8i+0.1j+0.2k& 0.2-0.6i+0.9j+0.1k\end{array}\right.\hfill \\ & \left.\begin{array}{cc}0.2i+0.3j+1.6k& -1-0.6i-0.5j-0.8k\\ -0.3-0.6i+0.2j-1k& 0.5-0.3i+0.5j+0.1k\\ 0.1-1.8i+j-0.7k& 1.4+1.2i-0.3j+1.2k\\ -0.3-1.2i-0.3j+0.1k& 0.2-0.6i+0.9j+0.1k\end{array}\right)ϵ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill E& =E_0+E_1ϵ,\hfill \\ \hfill E_0(:,1:2)& =\left(\begin{array}{cc}-28.376-26.517i-48.432j-3.783k& -12.196-21.227i-43.752j+12.387k\\ 18.501-25.157i-12.808j-5.144k& 5.231-23.747i-11.318j-13.664k\\ 20.626+6.097i+10.201j-11.1k& 19.386-2.953i+5.631j-9.68k\\ 2.911+7.635i+20.647j-10.44k& 9.481+7.975i+22.387j-10.86k\end{array}\right),\hfill \\ \hfill E_0(:,3:4)& =\left(\begin{array}{cc}-21.486-16.727i-32.122j+5.927k& -7.813+15.752i+32.997j-37.96k\\ 8.581-22.417i-11.248j-15.984k& -26.725-15.627i+4.871j-1.629k\\ 16.866+7.017i+6.981j-10.7k& 9.885-1.872i+12.539j+9.259k\\ 7.991+6.105i+18.797j-18.3k& 11.449-10.119i+4.697j+11.947k\end{array}\right),\hfill \\ \hfill E_1(:,1:2)& =\left(\begin{array}{cc}86.779-4.555i+18.066j+4.304k& 48.404-25.565i-19.591j+21.482k\\ 31.78+29.32i+27.871j-39.125k& 24.683+38.749i+11.953j-17.819k\\ -31.805+10.367i-26.219j+59.48k& -24.141-12.062i-42.234j+43.83k\\ -48.587-4.624i-27.528j+18.876k& -39.618-4.388i-15.191j+3.171k\end{array}\right),\hfill \\ \hfill E_1(:,3:4)& =\left(\begin{array}{cc}86.991-28.349i+10.429j-17.251k& -4.915-66.948i-4.951j-22.718k\\ 48.017+27.12i+27.542j-26.827k& 8.141-0.406i+55.097j+3.554k\\ -21.907+34.768i-11.6j+50.354k& -20.311+17.703i-42.871j-11.076k\\ -35.523-4.309i+1.102j-1.296k& -23.558+33.42i-5.801j+11.157k\end{array}\right).\hfill \end{array}$$

Then we examine the solvability of Equation (3). By computing with M, N, T, P, Q, $T_1$, $T_2$, $A_{11}$, and $B_{22}$ in Theorem 3.1, it can be calculated that

$$R_{T_1}{R}_{A_0}{E}_0{L}_{D_0}=0,R_{A_0}{E}_0{L}_{D_0}{L}_{T_2}=0,R_{M}N=0,R_{A_{11}}T{L}_{B_{22}}=0,Q{L}_P=0.$$

In this case, the Equation (3) is consistent. Using Matlab along with Algorithm 1, the general solution can be expressed as follows:

$$\begin{array}{cc}\hfill X_0& =G_{31}-A_0^{†}{V}_0{D}_0-G_{32}{V}_1{C}_0-G_{33}{V}_2{G}_{34}-G_{35}{V}_3{G}_{36}-G_{37}{V}_4{G}_{38}+G_{39}{V}_9+L_{A_0}{V}_{10}{G}_{40}\hfill \\ \hfill X_1& =G_{41}+G_{42}{V}_0{D}_0-A_0^{†}{V}_0{D}_1+G_{43}{V}_1{C}_0-G_{32}{V}_1{C}_1+G_{44}{V}_2{G}_{34}-G_{33}{V}_2{G}_{45}+G_{46}{V}_3{G}_{36}-G_{35}{V}_3{G}_{47}\hfill \\ \hfill & +G_{48}{V}_4{G}_{38}-G_{37}{V}_4{G}_{49}-G_{35}{V}_5{C}_0-G_{37}{V}_6{G}_{34}-G_{50}{V}_7{G}_{51}-A_0^{†}{V}_{11}{D}_0+L_{A_0}{V}_{12},\hfill \\ \hfill Y_0& =G_1+G_2{V}_1+G_3{V}_2{G}_4+G_5{V}_3{G}_6+V_4{G}_7,\hfill \\ \hfill Y_1& =G_8+G_5{V}_5+V_6{G}_4,\hfill \\ \hfill Z_0& =G_9+G_{10}{V}_0-G_{11}{V}_4{G}_{12}+G_{13}{V}_4{G}_{14}+G_{15}{V}_7{G}_{16},\hfill \\ \hfill Z_1& =G_{17}+G_{18}{V}_0{G}_{19}+G_{20}{V}_1{G}_{21}+G_{22}{V}_2{G}_{23}+G_{24}{V}_3{G}_{25}-G_{26}{V}_4{G}_{27}+G_{28}{V}_4{G}_{12}-G_{11}{V}_4{G}_{29}+G_{11}{V}_4{G}_{30}\hfill \\ \hfill & -G_{11}{V}_6{G}_{23}+G_{10}{V}_{11}+V_{13}{G}_{16}.\hfill \end{array}$$

(24)

For the sake of conciseness, the numerical values of $G_1,G_2,\dots ,G_{51}$ are not enumerated here. Specifically, if we select the random matrices $V_0,V_1,\dots ,V_7$ and $V_9,V_{10},\dots ,V_{13}$ in (24) as:

$$\begin{array}{cc}\hfill V_0& =\left(\begin{array}{cccc}2.5-i-2.5j-0.4k& -0.6-1.1i-2.2j-0.8k& -1.2-1.6i-1j+0.5k& -1+0.9i-j-0.4k\\ 2-0.3i-0.6j-1.6k& 0.4+0.3i+0.8j+0.6k& 0.6+0.7i+j+0.2k& -0.4-0.3i-0.2j-0.6k\\ -0.1+0.2i-0.4j-0.2k& 0.1+1.1i-0.6j-1k& 0.2-0.2i+0.7j-2.4k& -1.5+0.3i-0.5j-1.4k\\ 1.4+0.8i-1.1j-0.2k& 0.8+i+1.2j-0.2k& 0.3+1.4i+0.6j-0.8k& -0.7+0.5i-j\end{array}\right),\hfill \\ \hfill V_1& =\left(\begin{array}{cccc}-0.2-0.7i+0.7j-0.9k& -0.1-0.5i-2.2j+2.1k& -0.4-1.3i-0.2j-0.3k& 0.3-0.5i+1.4j-1.3k\\ 1.3+0.7i-0.3j+0.1k& -1.8+2.1i-0.1j-0.3k& 0.6+0.1i+1.2j+0.1k& -0.6-1.6i-0.2j-0.4k\\ 0.1+0.2i+1.4j+k& -1.5+0.1i+0.1j-0.2k& -0.8-0.4i-1.3j& -1.2+0.7i+0.3j+2.2k\\ 0.7+0.5i+0.3j+0.9k& -0.8-0.9i+0.6j-2.7k& -0.6-1.4i+2.5j-0.6k& -0.6+1.9i-0.9j-0.9k\end{array}\right),\hfill \\ \hfill V_2& =\left(\begin{array}{cccc}0.4i+1.3j+1.3k& 0.2-0.7i+1.2j+0.5k& 0.4i+0.6j+0.9k& 0.3-0.6i-0.1j+0.9k\\ -1.2-1.6i+0.7j-1.2k& 0.3i+0.4j+0.4k& 0.1-0.5i+1.2j-0.8k& 0.6+0.3i-j-0.3k\\ 1.2-0.7i+0.1j+0.3k& 1.5-0.2i+0.5j-0.9k& 1.5-2.1i-0.1j-1.2k& -0.1+i+0.2k\\ 0.8+1.1i-0.4j+0.9k& 0.6+0.3i+2.7j-0.1k& 0.5+i+1.8j-0.7k& -1-1.1i-1.7j-1.7k\end{array}\right),\hfill \\ \hfill V_3& =\left(\begin{array}{cccc}-0.8+2.2i+1.7j-k& 1-1.3i-0.5j+0.6k& -0.2+0.7i+2.2j+k& -0.7+0.5j+0.6k\\ -0.7-0.2i+0.8j+0.1k& -1.6-0.4i-0.6j+0.1k& 0.6i+0.9j-0.7k& 0.8-1.4i+0.7j\\ -1.3-0.5i+j-1.5k& 0.2-0.5i+1.1j-0.9k& 1.7+0.9i-0.6j-1.3k& 0.9-1.7i+1.2j+0.3k\\ 1.2+0.5i+1.8j+0.6k& -1+1.4i-0.7j+1.3k& 0.4+0.9i-0.4j+k& 0.1+0.3j+0.7k\end{array}\right),\hfill \\ \hfill V_4& =\left(\begin{array}{cccc}-0.4+0.2i+0.6j+0.5k& -0.5-2.2j-0.2k& 1.2+0.6i+0.6j+0.5k& 0.6-1.1i+1.8j+0.2k\\ 1.6+0.8i-1.6j-1.2k& 1.5-2.1i-0.2j+0.7k& 0.3+0.7i+1.1j-0.5k& -0.1i-0.1j-0.6k\\ 0.3+0.3j& -0.6+1.2i-1.3j+2k& 2.2+0.3i-0.2j+0.1k& 0.4+1.1i+0.4j+k\\ -0.8-0.4i+1.1j-0.9k& -1.1+0.3i-j-0.5k& -0.3+1.4i-0.5j+2.8k& -1.6-0.3j+0.9k\end{array}\right),\hfill \\ \hfill V_5& =\left(\begin{array}{cccc}-1.5-0.3i+0.3j-0.8k& 0.1+1.8i-1.1j+0.4k& -0.8+0.1i-0.9j-k& -0.2-0.1i-0.6j-0.9k\\ -0.6-0.1i-1.4j+k& -0.6+0.9i-0.2j-1.2k& 0.4-1i+0.8j+0.5k& 0.7+1.2i-0.1j-1.1k\\ -0.4-0.3i-0.5j+1.7k& 0.4-1j+0.7k& -1.1-0.1i+1.3k& -1-0.5i+0.8j-1.2k\\ -0.3+0.7i+j-0.9k& 0.5-0.7i-j+0.1k& 0.8+0.9i+0.4j-0.2k& 0.8+0.4i-1.3j+0.4k\end{array}\right),\hfill \\ \hfill V_6& =\left(\begin{array}{cccc}0.3+1.4i-0.4k& 1.5-0.1i+0.2j+0.6k& -1.4i-0.3j-1.2k& -0.5+0.4i-0.3j+0.6k\\ -0.8+0.4i+0.4j-0.2k& -2.1+0.6i+1.1j+0.1k& -1.5i-0.5j-2.2k& 0.7+1.5i+0.9k\\ 0.8-1.5i+0.1j-1.1k& -0.3+0.7i-2.7j+1.4k& -0.5+1.3i+0.1j+0.1k& 1.2+1.8i-1.6j-0.2k\\ -1.1-0.5i+0.2j-1.2k& 0.8+0.7i+0.5k& 2+0.8i+0.5j-0.2k& -0.3-0.8i-1.1j+0.1k\end{array}\right),\hfill \\ \hfill V_7& =\left(\begin{array}{cccc}-0.7+1.2i-1.6j-0.5k& -0.5+0.2i+0.2j+1.4k& 0.3+0.1i-3.1j+0.7k& 1.7-0.7i+1.1j+0.4k\\ 1.2+0.8i-0.5j-1.2k& 1.8-1.8i-0.4j-0.1k& 0.5+i-0.9j-0.5k& -0.2-0.5i+0.5j-0.3k\\ 0.1+1.2i+2.9j+0.4k& 0.5+0.9i-1.8j-1.5k& 0.5-0.3i-1.5j+0.6k& -0.8+0.5i-0.4j-0.1k\\ -0.5-0.2i+0.5j-0.6k& 0.4-0.1i+0.1j+0.1k& 0.4i-0.2j+0.3k& -0.4-0.2i-1.6j-0.1k\end{array}\right),\hfill \\ \hfill V_9& =\left(\begin{array}{cccc}-1.5-0.1i+0.6j-0.5k& -1.5i-1.8j+0.7k& 0.2+0.3i+1.1j-0.4k& -0.6-0.6i+0.6j-0.3k\\ -0.1+0.1i+2.1j+0.3k& -0.8-0.1i+0.1j-0.5k& 0.3+0.9i-0.2j+0.5k& 0.4+0.4i+0.3j-1.1k\\ -0.8-1.1i-1.1j-0.4k& 1.3+0.2i+0.8j+0.4k& -2-1.6i+0.2j+0.3k& -0.2+0.3i+1.4j+0.1k\\ -1.3+0.1i-0.8j-0.6k& 2.2-0.3i-0.4j-0.2k& 1.5-1.5i+j-2k& 0.3-0.4i-0.9j-0.7k\end{array}\right),\hfill \\ \hfill V_{10}& =\left(\begin{array}{cccc}-0.7i+0.6j+1.5k& -1.6-1.4i-0.5j+0.4k& 2.4-1.7i+0.5j+1.1k& -0.4+1.5i+1.2j+0.5k\\ 1.3-1.4i+0.8j+0.3k& 1.5+0.7i+0.6j+0.4k& -0.5-0.5i-0.4j-2.5k& -1.2-0.4i-0.9j-0.3k\\ 1.7-2.7i-1.5j-0.4k& 1.1+1.1i+1.1j-0.2k& 1.8+0.9i-0.5j-k& 0.6-0.7i-1.9j\\ 0.8-i-0.1j+0.2k& 0.3-2i+1.2j+2.3k& -1.1+1.3i-1.2j& 0.4-i-0.8j-0.3k\end{array}\right),\hfill \\ \hfill V_{11}& =\left(\begin{array}{cccc}-1.3-1.6i+0.5j-1.5k& -0.9-1.1i-0.9j+0.2k& 0.9-1.4i+1.1j+0.6k& -0.4+0.7i+0.4j+0.5k\\ 2.1+0.3i-0.1j-0.5k& -0.3+1.5i+0.4j+0.3k& 0.8-1i+1.4j-1.2k& 0.8+1.2i-1.6j-0.5k\\ -0.2-1.5i+1.3k& -1.3-0.2i+1.6j+1.8k& -0.9+0.3i+1.1j-0.9k& -0.8-0.9i-0.8j+0.3k\\ 2.3-0.7i-0.9j-0.8k& -0.9+0.3i-0.3j-0.3k& 0.2-0.1i+1.5j+0.3k& 1.6-i-0.2j-0.7k\end{array}\right),\hfill \\ \hfill V_{12}& =\left(\begin{array}{cccc}0.6-0.2i+0.5j-0.4k& 1.4-0.8i+0.7j+1.1k& 0.1+2i-0.1j+0.4k& 1.2-0.9i-2.2j-0.4k\\ -0.7-0.9i-0.2j+0.3k& -1.1-0.6i-0.6j-0.1k& 0.8+0.4i-0.9j-0.3k& 1.7+0.2i+0.1j+1.6k\\ 0.3-0.1i+0.3j-0.6k& 0.2+1.4i+0.4j-1.1k& -1.1+0.4i+0.2j+1.7k& -0.4i+0.1j+0.3k\\ 0.4-1.7i-0.5j+0.3k& -1.7+0.2i+0.7j-0.1k& -0.2-0.2i+1.1j-0.4k& 1.5-1.6i+0.3j+0.5k\end{array}\right),\hfill \\ \hfill V_{13}& =\left(\begin{array}{cccc}1.3-1.1i-0.3j-0.4k& -0.8-1.7i-2.2j+0.8k& 2.2+0.5i+0.4j+0.4k& -0.9+0.6i-0.3j+0.8k\\ -0.7-i+1.9j+0.5k& 0.2-1.7i-1.5j-0.3k& -0.2+0.7i+2.5j+1.1k& -0.4-0.1i-0.9j-0.5k\\ -0.2+0.3i-0.2j+0.2k& -1.2-1.7i-0.8j-0.7k& -0.4-1.1i-1.8j+0.1k& -j+1.8k\\ 0.9-0.2i+0.5j-1.6k& 0.5+0.1j-0.2k& 1.7-2i-1.7j-1.7k& 0.6-0.4i+j+0.6k\end{array}\right),\hfill \end{array}$$

thus, a specific solution of Equation (3) is

$$\begin{array}{cc}\hfill X_{00}(:,1:2)& =\left(\begin{array}{ccc}1.162862-7.41657i+1.832169j-5.282234k& -1.051602-3.938339i+4.257563j-6.991917k& \\ -5.435692+6.505646i-1.187703j+4.749774k& -4.465519+5.358167i+0.191202j+5.493997k\\ 1.012862-6.11657i+1.982169j-7.382234k& 0.348398-3.138339i+3.257563j-7.291917k\\ -4.785692+7.605646i+0.862297j+6.849774k& -8.665519+2.858167i+1.291202j+7.093997k\end{array}\right),\hfill \\ \hfill X_{00}(:,3:4)& =\left(\begin{array}{cc}-2.137138-7.21657i+2.682169j-3.732234k& -3.251633-1.285817i+12.915869j+1.608638k\\ -4.385692+3.605646i+2.662297j+3.799774k& 1.925612+6.419289i-9.292437j+2.161586k\\ 0.612862-5.01657i+2.032169j-6.432234k& -2.651633-4.385817i+9.015869j+0.708638k\\ -6.135692+7.105646i+0.612297j+7.499774k& 3.625612+6.619289i-7.992437j+1.761586k\end{array}\right),\hfill \\ \hfill X_{01}(:,1:2)& =\left(\begin{array}{cc}5.814901+4.084309i+15.638418j+7.941647k& 1.09511-0.179288i+7.803014j+11.634892k\\ -3.808465-9.506473i-13.140385j+1.972355k& 0.240757-8.473277i-5.328949j+0.095532k\\ 5.514901+4.184309i+15.438418j+7.741647k& -0.10489+2.020712i+7.503014j+9.434892k\\ -2.708465-10.306473i-13.440385j+1.972355k& -0.359243-7.673277i-4.028949j+0.095532k\end{array}\right),\hfill \\ \hfill X_{01}(:,3:4)& =\left(\begin{array}{cc}7.574157+7.807107i+11.072523j+1.574317k& 0.320117+3.20723i-6.746305j+17.445775k\\ -11.264143-8.766344i-12.197597j+6.4044k& 2.351403-4.210274i-7.849136j-7.609446k\\ 6.374157+6.207107i+11.372523j+2.874317k& -0.879883+3.70723i-4.446305j+18.145775k\\ -12.264143-9.366344i-10.197597j+6.3044k& 2.151403-6.010274i-7.649136j-8.709446k\end{array}\right),\hfill \\ \hfill Y_{00}(:,1:2)& =\left(\begin{array}{cc}-1.217003+0.583393i+1.581191j-0.750215k& 0.450598-0.849911i-4.103562j+0.041317k\\ 1.6187-0.664884i-1.504164j-1.671444k& -1.624933-0.177218i-1.2887j+0.519676k\\ -0.907998-0.386761i+0.904412j-0.234593k& 0.994836+0.820616i-1.837318j+1.501705k\\ 2.5187+1.335116i+1.695836j+2.028556k& -2.024933+1.022782i+0.8113j-2.380324k\end{array}\right),\hfill \\ \hfill Y_{00}(:,3:4)& =\left(\begin{array}{cc}-0.406875+0.516895i-0.802389j+0.381489k& 0.837866-1.938093i+3.037137j+1.444556k\\ 0.692355+0.730691i+3.089026j-1.918138k& 1.738851-2.512174i-0.913586j-1.170942k\\ 1.908611-0.712376i-1.468862j-0.137159k& 0.645233+1.493394i+0.820374j+2.392911k\\ -0.307645+1.730691i+2.089026j+2.481862k& -2.161149+1.087826i-2.913586j-0.870942k\end{array}\right),\hfill \\ \hfill Y_{01}(:,1:2)& =\left(\begin{array}{cc}0.018635+1.141622i+0.055493j-0.41456k& 1.610536-0.044855i-0.041071j+1.13057k\\ -0.622818+0.112415i-0.705911j+0.486487k& -3.104348+1.059492i+1.473183j-0.934379k\\ 0.731511-1.737086i+0.08251j-0.928413k& -0.0608+0.921522i-3.001536j+1.535554k\\ -0.622818+0.012415i+1.494089j-2.413513k& 0.895652-0.440508i-0.426817j+0.765621k\end{array}\right),\hfill \\ \hfill Y_{01}(:,3:4)& =\left(\begin{array}{cc}-0.805546-1.315357i-0.268597j-1.541675k& -0.549219-0.133397i-0.038682j-0.115294k\\ 0.517177-2.475962i+0.41198j-1.781291k& 1.260781+2.750919i+0.458721j+0.33183k\\ -1.033313+0.906353i-0.073319j+0.074355k& 1.132467+1.289777i-1.09136j-0.254915k\\ 2.917177+1.724038i+1.01198j-0.481291k& 0.360781-0.349081i-1.841279j+1.03183k\end{array}\right).\hfill \\ \hfill Z_{00}(:,1:2)& =\left(\begin{array}{cc}3.656028-4.782481i-3.462455j+3.46199k& 6.148981-3.659563i+3.767115j-2.893348k\\ 5.47224-5.156069i-9.57172j-9.13546k& 11.288362+8.742963i-4.364158j+3.720651k\\ -2.003439-3.703678i+13.950491j-4.036224k& -7.598024-0.13754i-7.546665j-22.925728k\\ -7.076557-0.163186i+0.797067j-7.720596k& -0.700287+2.134835i-11.720979j-0.059761k\end{array}\right),\hfill \\ \hfill Z_{00}(:,3:4)& =\left(\begin{array}{cc}-2.688134+2.709677i-2.239314j+6.454926k& -7.966349+0.378787i-3.549527j-6.791858k\\ -0.118774-8.23759i-3.011795j+5.575336k& -5.095572+5.256642i+14.176442j-1.362569k\\ 17.820611-2.206649i+5.209183j+3.199725k& -12.265339+14.443124i-6.141329j+17.988096k\\ -2.433648-6.573347i+3.105061j-8.776877k& 9.351545+3.099762i+1.95509j+7.75759k\end{array}\right),\hfill \\ \hfill Z_{01}(:,1:2)& =\left(\begin{array}{cc}-1.746371-2.520745i+2.543229j-4.285164k& 0.824995-1.997068i+1.233662j+0.311536k\\ 5.288869+7.575129i+11.406692j+2.466337k& -5.032716+7.746839i-0.610895j+0.891857k\\ -1.020457+2.157938i-5.57635j-1.004305k& 6.244273-3.060239i+2.130841j-5.894446k\\ 9.946278+6.250725i-14.216645j+2.081148k& 1.982303+0.261978i-10.196027j+12.94616k\end{array}\right),\hfill \\ \hfill Z_{01}(:,3:4)& =\left(\begin{array}{cc}5.246913+2.443496i+2.744987j+2.082108k& 0.276575+0.924436i+1.349027j-1.116736k\\ 0.095025+0.40358i+1.296004j+6.06396k& 4.505386-5.638816i-1.656447j-9.774048k\\ 7.552789-6.188465i-6.918277j-7.030933k& -8.883547+10.68105i-7.567458j-4.825046k\\ 1.054306-2.634416i-3.821167j-1.174813k& -4.472455+10.948272i+13.784792j+0.246788k\end{array}\right).\hfill \end{array}$$

Therefore, we obtain

$$\begin{array}{cc}\hfill & \parallel A_0{X}_{00}+B_0{Y}_{00}{C}_0+Z_{00}{D}_0-E_0{\parallel }_F=2.5598\times {10}^{-13},\hfill \\ \hfill & \parallel A_1{X}_{00}+A_0{X}_{01}+B_1{Y}_{00}{C}_0+B_0{Y}_{01}{C}_0+B_0{Y}_{00}{C}_1+Z_{00}{D}_1+Z_{01}{D}_0-E_1{\parallel }_F=3.0652\times {10}^{-13}.\hfill \end{array}$$

The Frobenius norm is sensitive to every element of the matrix, so any modification may change its value. Since we naturally desire a more accurate reflection of the error, the Frobenius norm is adopted for error computation.

Additionally, we randomly generated 10,000 different sets of $V_0,V_1,\dots ,V_7$ and $V_9,V_{10},\dots ,V_{13}$ , all resulting errors were less than $5\times {10}^{-13}$. Hence, Theorem 3.1 is effective for solving the dual quaternion matrix equation (3) within the permissible error range.

5. Applications

In this section, we present an image encryption and decryption scheme based on matrix equation (3), which is successfully validated through examples. A quaternion matrix is capable of representing a single colour image, while a dual quaternion matrix, which contains two quaternion matrices, can represent two colour images simultaneously, thereby making it more attractive in the field of image processing. In the proof of Theorem 3.1, it is known that equation (3) is equivalent to system (9). Our encryption and decryption procedures are carried out based on system (9), as shown in Figure 1 .

Now, we describe the overall framework for encryption and decryption. In matrix system (9), two equations involve six variables. By imposing $X_1=X_0$, $Y_1=Y_0$, $Z_1=Z_0$ and treating the image represented by $Z_0$ as the key image, encryption and decryption procedures are facilitated with enhanced smoothness and efficiency. Figure 1 is composed of the following components. $A_0$ ,$A_1$,$B_0$,$B_1$,$C_0$,$C_1$,$D_0$ and $D_1$ are encryption matrices, or the coefficient matrices. $X_0$ and$Y_0$ denote two original images, $Z_0$ is a key image. These images are encrypted using the formulas presented in Figure 1, yielding two encrypted images $E_0$ and $E_1$. These encryption matrices are publicly accessible, only the receiver holds the key image , the receiver can restore original images from their encrypted counterparts. The detailed encryption and decryption processes are provided by Algorithm 2 and 3.

Algorithm 2 Encryption Process

1. Input: Two original colour images , one key image and encryption matrices .

2. Let $X_0$ and $Y_0$ represent two colour images respectively. $Z_0$ represents the key image .

The encryption matrices are $A_0$ ,$A_1$,$B_0$,$B_1$,$C_0$,$C_1$,$D_0$ and $D_1$ .

3. Encrypt $X_0$ and $Y_0$ using the encryption method in Figure 1.

4.Two encrypted images, $E_0$ and $E_1$, are obtained.

5. Output: Two encrypted colour images.

Algorithm 3 Decryption Process

1. Input: Two encrypted colour images , one key image and encryption matrices.

2. $E_0$ and $E_1$ are two encrypted colour images.

3. By applying Theorem 3.1 and setting $X_1=X_0$, $Y_1=Y_0$,$Z_1=Z_0$, the matrices $X_0$ and $Y_0$

can be derived.

4. Two colour images, denoted as $X_0$ and $Y_0$, can be obtained.

5. Output: Two decrypted colour images .

Using two original images and one as the key, Algorithms 2 and 3 are applied so that the results shown in Figure 2 are obtained. To evaluate the performance,the Structural Similarity Index (SSIM), Feature Similarity Index (FSIM) and Peak Signal-to-Noise Ratio (PSNR) are computed, the results are recorded in Table 4.

From Table 4, it can be observed that the SSIM and FSIM values both equal 1, which implies perfect decryption with negligible information loss. In addition, the PSNR values consistently exceed 42, indicating outstanding image quality and a high degree of similarity to the original images. The results presented in Table 4 confirm the feasibility of the proposed scheme.

6. Conclusions

In this paper, we derive the necessary and sufficient conditions for the solvability of the dual quaternion matrix equation (3) by adopting the methods of generalized inverses and matrix rank equalities, and present the general solutions expression when the equation is solvable. Afterwards, we give a numerical example to verify the theoretical results. Finally, the application to color image encryption and decryption demonstrates that the model characterized in Figure 1 achieves satisfactory performance. Future work may involve further investigation of matrix equations over commutative quaternions.