Constrained Solutions of a System of Matrix Equations

Abstract

We derive the necessary and sufficient conditions of and the expressions for the orthogonal solutions, the symmetric orthogonal solutions, and the skew-symmetric orthogonal solutions of the system of matrix equations $AX=B$ and $XC=D$, respectively. When the matrix equations are not consistent, the least squares symmetric orthogonal solutions and the least squares skew-symmetric orthogonal solutions are respectively given. As an auxiliary, an algorithm is provided to compute the least squares symmetric orthogonal solutions, and meanwhile an example is presented to show that it is reasonable.