# Generalization of Roth's solvability criteria to systems of matrix equations

## TL;DR

This paper extends Roth's classical solvability criterion for matrix equations to complex and quaternion matrix systems, providing new conditions for the existence of solutions in these broader contexts.

## Contribution

The paper generalizes Roth's criterion to systems involving complex conjugation and quaternion matrices, broadening the applicability of solvability conditions for matrix equations.

## Key findings

- Extended Roth's criterion to complex conjugate matrix systems
- Established solvability conditions for quaternion matrix equations
- Unified framework for real, complex, and quaternion matrix equations

## Abstract

W.E. Roth (1952) proved that the matrix equation $AX-XB=C$ has a solution if and only if the matrices $\left[\begin{matrix}A&C\\0&B\end{matrix}\right]$ and $\left[\begin{matrix}A&0\\0&B\end{matrix}\right]$ are similar. A. Dmytryshyn and B. K{\aa}gstr\"om (2015) extended Roth's criterion to systems of matrix equations $A_iX_{i'}M_i-N_iX_{i''}^{\sigma_i} B_i=C_i$ $(i=1,\dots,s)$ with unknown matrices $X_1,\dots,X_t$, in which every $X^{\sigma}$ is $X$, $X^T$, or $X^*$. We extend their criterion to systems of complex matrix equations that include the complex conjugation of unknown matrices. We also prove an analogous criterion for systems of quaternion matrix equations.