The general $\phi$-Hermitian solution to mixed pairs of quaternion matrix Sylvester equations

Main Article Content

Zhuo-Heng He
Jianzhen Liu
Tin-Yau Tam

Abstract

Let $\mathbb{H}^{m\times n}$ be the space of $m\times n$ matrices over $\mathbb{H}$, where $\mathbb{H}$ is the real quaternion algebra. Let $A_{\phi}$ be the $n\times m$ matrix obtained by applying $\phi$ entrywise to the transposed matrix $A^{T}$, where $A\in\mathbb{H}^{m\times n}$ and $\phi$ is a nonstandard involution of $\mathbb{H}$. In this paper, some properties of the Moore-Penrose inverse of the quaternion matrix $A_{\phi}$ are given. Two systems of mixed pairs of quaternion matrix Sylvester equations $A_{1}X-YB_{1}=C_{1},~A_{2}Z-YB_{2}=C_{2}$ and $A_{1}X-YB_{1}=C_{1},~A_{2}Y-ZB_{2}=C_{2}$ are considered, where $Z$ is $\phi$-Hermitian. Some practical necessary and sufficient conditions for the existence of a solution $(X,Y,Z)$ to those systems in terms of the ranks and Moore-Penrose inverses of the given coefficient matrices are presented. Moreover, the general solutions to these systems are explicitly given when they are solvable. Some numerical examples are provided to illustrate the main results.

Article Details

Section
Article