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

W.E. Roth (1952) proved that the matrix equation AX−XB=C has a solution if and only if the matrices [AC0B] and [A00B] are similar. A. Dmytryshyn and B. KÃ¥gström (2015) extended Roth's criterion to systems of matrix equations AiXi′Mi−NiXi″σiBi=Ci (i=1,…,s) with unknown matrices X1,…,Xt, in which every Xσ is X, X⊤, 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.