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The matrix equation $AX-XB=C$ has a solution if and only if the matrices[A&C\\0&B] and [A &0\\0 & B] are similar. This criterion was proved over afield by W.E. Roth (1952) and over the skew field of quaternions by HuangLiping (1996). H.K. Wimmer (1988) obtained an analogous criterion for thematrix equation $X-AXB=C$ over a field. We extend these criteria to the matrixequations $AX-\widehat XB=C$ and $X-A\widehat XB=C$ over the skew field ofquaternions with a fixed involutive automorphism $q\mapsto \hat q$.

linear algebra and its applications

Roth's solvability criteria for the matrix equations ${AX-\widehat  XB=C}$ and ${X-A\widehat{X}B=C}$ over the skew field of quaternions with an  involutive automorphism $q\mapsto \hat q$