The Solvability of a System of Quaternion Matrix Equations Involving ϕ-Skew-Hermicity

Symmetry

Zhuo-Heng He¹  Xiao-Na Zhang¹  Yun-Fan Zhao²  Shao-Wen Yu³ 

[1] Department of Mathematics, Shanghai University, Shanghai 200444, China;Department of Mathematics, University of California, Los Angeles, CA 90095, USA;School of Mathematics, East China University of Science and Technology, Shanghai 200237, China;

关键词: quaternion algebra;    matrix decompositions;    matrix equations;    ϕ-skew-Hermicity;   

DOI  :  10.3390/sym14061273

来源: DOAJ

【 摘 要 】

Let $\mathbb{H}$ be the real quaternion algebra and ${\mathbb{H}}^{m\times n}$ denote the set of all $m\times n$ matrices over $\mathbb{H}$. For $A\in {\mathbb{H}}^{m\times n},$ we denote by $A_{\varphi }$ the $n\times m$ matrix obtained by applying $\varphi$ entrywise to the transposed matrix $A^T,$ where $\varphi$ is a non-standard involution of $\mathbb{H}$. $A\in {\mathbb{H}}^{n\times n}$ is said to be $\varphi$-skew-Hermicity if $A=-A_{\varphi }$. In this paper, we provide some necessary and sufficient conditions for the existence of a $\varphi$-skew-Hermitian solution to the system of quaternion matrix equations with four unknowns $A_i{X}_i{\left(A_i\right)}_{\varphi }+B_i{X}_{i+1}{\left(B_i\right)}_{\varphi }=C_i,(i=1,2,3),A_4{X}_4{\left(A_4\right)}_{\varphi }=C_4$.

【 授权许可】

Unknown