On the solvability for the mixed-type Lyapunov equation

摘要

In this paper, the linear matrix equation X = AXB(*) + BXA(*) + Q is considered, which is called the mixed-type Lyapunov equation. Some necessary and sufficient conditions for the existence of a unique solution are presented. Since a Hermitian positive semidefinite solution is important from the application point of view, some sufficient conditions for the existence of a Hermitian positive semidefinite solution are derived.