The $\eta$-Hermitian solutions of some quaternion matrix equations

Document Type : Original Article

Authors

Radja Belkhiri

Sihem Guerarra

Department of Mathematics and Informatics, Dynamic Systems and Control Laboratory, Faculty of Exact Sciences and Sciences of Nature and Life, University of Oum El Bouaghi, 04000, Algeria.

10.22034/kjm.2024.392808.2836

Abstract

Let $\mathbb{H}^{n\times m}$ be the set of all $n\times m$ matrices over the real quaternion algebra. In this paper, we derive the solvability conditions for the common $\eta$-Hermitian solution to the system of two quaternion matrix equations $A_{1}X_{1}A_{1}^{\eta\ast}+B_{1}Y_{1}B_{1}^{\eta\ast}=C_{1}$ and $A_{2}X_{2}A_{2}^{\eta\ast}+B_{2}Y_{2}B_{2}^{\eta\ast}=C_{2}$. As applications, we obtain necessary and sufficient conditions for the pair of quaternion matrix equations $A_{1}X_{1}A_{1}^{\eta\ast}=C_{1}$ and $A_{2} X_{2}A_{2}^{\eta\ast}=C_{2}$ to have common $\eta$-Hermitian solution. In additions, we establish formulas of the extremal ranks of the quaternion $\eta$-Hermitian matrix expression $A_{2}X_{2}A_{2}^{\eta\ast}=C_{2}$ with respect to $\eta$-Hermitian solution of $A_{1}X_{1}A_{1}^{\eta\ast}=C_{1}$, then we derive extremal ranks of the generalized $\eta$-Hermitian Schur complement $S_{A_{1}}=D-B^{\eta\ast}A_{1}^{-}B$ with respect to $\eta$-Hermitian generalized inverse $A_{1}^{-}$ of $A_{1}$, which is a solution to the quaternion matrix equation $A_{1}X_{1}A_{1}^{\eta\ast}=C_{1}$.

Keywords

System of quaternion matrix equations

Moore-Penrose inverse

$η$-Hermitian solution

Schur Complement

Subjects

• 15 Linear and multilinear algebra; matrix theory

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