The matrix equations AX=B,XC=D with PX=sXP constraint

摘要

The matrix equations AX=B,XC=D with the constraint PX=sXP are considered, where P is a given Hermitian matrix satisfying P2=I and s=±1. Using an eigenvalue decomposition of P, the constrained problem can be equivalently transformed to two independent (unconstrained) problems of matrix equations in similar forms, and hence the constrained problem can be solved in terms of the eigenvectors of P. In this paper, a simple and eigenvector-free formula of the general solutions to the constrained problem is presented using Moore–Penrose generalized inverses of the coefficient matrices A and C. A similar problem of the matrix equations with generalized constraint is discussed, too. We also consider the least-norm solution of the constrained problems.