The solutions of the Sylvester-like quaternion matrix equation $AX^{\varepsilon}+X^{\delta}B=0$

Abstract

In this paper, we discuss the Sylvester-like quaternion matrix equation $AX^\varepsilon+X^\delta B=0$, where $\varepsilon\in{\mathbb{I},\mathbb{C}\}$,$\delta\in\{\dag,*\}$ and $\mathbb{I}$,$\mathbb{C}$,$\dag$,$*$ denote the identity mapping, involutive automorphism, involutive anti-automorphism and transpose, involutive automorphism and anti-automorphism and transpose, respectively. Firstly, we transform the given equation into the new equation $\widetilde{A}Y^\varepsilon+Y^\delta\widetilde{B}=0$ with complex coefficient matrices $\widetilde{A},\widetilde{B}$ and unknown quaternion matrix $Y$ by utilizing the regularity of the matrix pencil $(A,B^{\varepsilon\delta})$, where $\widetilde{A}=PAQ$ and $\widetilde{B}=Q^{\delta\varepsilon}BP^{\delta\varepsilon}$ with $P,Q$ being two nonsingular quaternion matrices. Secondly, we decouple the transformed equation into some systems of small-scale equations in terms of Kronecker canonical form of $(\widetilde{A},\widetilde{B}^{\varepsilon\delta})$. Moreover, we also show that the solution can be gotten in terms of $P,Q,$ the Kronecker canonical form of $(\widetilde{A},\widetilde{B}^{\varepsilon\delta})$ and the two nonsingular quaternion matrices which transform $(\widetilde{A},\widetilde{B}^{\varepsilon\delta})$ into its Kronecker canonical form. Thirdly, we determine the dimension of the solution space of the equation in terms of the sizes of the blocks arising in the Kronecker canonical form. Finally, we give the necessary and sufficient condition for the existence of the unique solution.