# Relation between the T-congruence Sylvester equation and the generalized Sylvester equation

## TL;DR

This paper explores the relationship between the T-congruence Sylvester equation and the generalized Sylvester equation, providing new transformations for rectangular matrices to extend previous results beyond square matrices.

## Contribution

The paper introduces two new transformations that relate the T-congruence Sylvester equation to the generalized Sylvester equation for rectangular matrices, extending prior work limited to square matrices.

## Key findings

- Provided a transformation for the case m ≥ n
- Developed a novel transformation for m ≤ n
- Extended the applicability of existing transformations

## Abstract

The T-congruence Sylvester equation is the matrix equation $AX+X^{\mathrm{T}}B=C$, 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 $X\in\mathbb{R}^{n\times m}$ is to be determined. Recently, Oozawa et al. discovered a transformation that the matrix equation is equivalent to one of the well-studied matrix equations (the Lyapunov equation); however, the condition of the transformation seems to be too limited because matrices $A$ and $B$ are assumed to be square matrices ($m=n$). In this paper, two transformations are provided for rectangular matrices $A$ and $B$. One of them is an extension of the result of Oozawa et al. for the case $m\ge n$, and the other is a novel transformation for the case $m\le n$.