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

Abstract EN

The matrix equation AXBH=E with SX=XR or PX=sXQ constraint is considered, where S, R are Hermitian idempotent, P, Q are Hermitian involutory, and s=±1.

By the eigenvalue decompositions of S, R, the equation AXBH=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 AXBH=E with PX=sXQ constraint.