  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.

Representation Theory