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

TL;DR

This paper explores the connection between the T-congruence Sylvester equation and the Lyapunov equation, showing that under certain conditions, the former can be transformed into the latter, enabling the use of existing solution methods.

Contribution

It establishes a transformation from the T-congruence Sylvester equation to the Lyapunov equation under specific conditions, facilitating new analytical and numerical approaches.

Findings

Transformation from T-congruence Sylvester to Lyapunov equation under certain conditions

Potential for applying Lyapunov equation solution techniques to T-congruence Sylvester problems

Provides theoretical foundation for efficient numerical solvers

Abstract

We consider the T-congruence Sylvester equation $AX+X^{\rm T}B=C$, where $A\in \mathbb R^{m\times n}$, $B\in \mathbb R^{n\times m}$ and $C\in \mathbb R^{m\times m}$ are given, and matrix $X \in \mathbb R^{n\times m}$ 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 note, we will show that, under a certain condition, the T-congruence Sylvester equation can be transformed into the Lyapunov equation. This may lead to further properties and efficient numerical solvers by utilizing a great deal of studies on the Lyapunov equation.