Eigenvector-Free Solutions to the Matrix Equation AXBH=E with Two Special Constraints

Abstract

The matrix equation $AX{B}^H=E$ with $SX=XR$ or $PX=sXQ$ constraint is considered, where S, R are Hermitian idempotent, P, Q are Hermitian involutory, and $s=\pm 1$. By the eigenvalue decompositions of S, R, the equation $AX{B}^H=E$ with $SX=XR$ constraint is equivalently transformed to an unconstrained problem whose coefficient matrices contain the corresponding eigenvectors, with which the constrained solutions are constructed. The involved eigenvectors are released by Moore-Penrose generalized inverses, and the eigenvector-free formulas of the general solutions are presented. By choosing suitable matrices S, R, we also present the eigenvector-free formulas of the general solutions to the matrix equation $AX{B}^H=E$ with $PX=sXQ$ constraint.