On a relationship between the T-congruence Sylvester equation and the Lyapunov equation

The T-congruence Sylvester equation is the matrix equation AX+XTB=C, where A∈Rm×n, B∈Rn×m and C∈Rm×m are given, and matrix X∈Rn×m is to be determined. The T-congruence Sylvester equation has recently attracted attention because of a relationship with palindromic eigenvalue problems. For example, necessary and sufficient conditions for the existence and uniqueness of solutions, and numerical solvers have been intensively studied. In this paper, we will show that, under a certain condition and n=m, the T-congruence Sylvester equation can be transformed into the Lyapunov equation. This may lead to further properties and efficient numerical solvers by utilizing the rich literature on the Lyapunov equation.