On Solutions of Matrix Equation $AXA^T+BYB^T=C$

Abstract

By making use of the quotient singular value decomposition (QSVD) of a matrix pair, this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the general solutions of the linear matrix equation $AXA^T+BYB^T=C$ with the unknown $X$ and $Y$, which may be both symmetric, skew-symmetric, nonnegative definite , positive definite or some cross combinations respectively. Also, the solutions of some optimal problems are derived.