The least squares $\eta$-Hermitian problems of quaternion matrix equation $A^HXA+B^HYB=C$

Abstract

For any $A=A_1+A_2j\in \mathbf{Q}^{n\times n}$ and $\eta \in\{i,j,k\},$ denote $A^{\eta H}=-\eta A^H\eta.$ If $A^{\eta H}=A,$ $A$ is called an $\eta$-Hermitian matrix. If $A^{\eta H}=-A,$ $A$ is called an $\eta$-anti-Hermitian matrix. Denote $\eta$-Hermitian matrices and $\eta$-anti -Hermitian matrices by $\mathbf{\eta HQ}^{n\times n}$ and $ \mathbf{\eta
AQ}^{n\times n},$ respectively.

In this paper, we consider the least squares $\eta$-Hermitian problems of quaternion matrix equation $A^HXA+B^HYB=C$ by using the complex representation of quaternion matrices, the
Moore--Penrose generalized inverse and the Kronecker product of matrices. We derive the expressions of the least squares solution with the least norm of quaternion matrix equation $A^HXA+B^HYB=C$ over $[X, Y]\in \mathbf{\eta HQ}^{n\times n}\times \mathbf{\eta HQ}^{k\times k},$ $[X, Y]\in \mathbf{\eta AQ}^{n\times n}\times \mathbf{\eta
AQ}^{k\times k},$ and $[X, Y]\in \mathbf{\eta HQ}^{n\times n}\times \mathbf{\eta
AQ}^{k\times k},$ respectively.