The symmetric solution of the matrix equations AX + YA = C, AXA T + BYB T = C, and ( A TXA , B TXB ) = ( C, D)

Abstract

The necessary and sufficient conditions for the existence of and the expressions for the symmetric solutions of matrix equations (I) AX + YA = C, (II) AXA T + BYB T = C, and (III) ( A TXA , B TXB ) = ( C, D) are derived. In addition, the minimum-2-norm least-squares symmetric solution of equation (I), the minimum-2-norm symmetric solution of equation (II), and the least-squares solution of equation (III) are obtained.