The Matrix Equation $A\bar X - XB = C$ and Its Special Cases
Jean H. Bevis, Frank J. Hall, Robert E. Hartwig · SIAM Journal on Matrix Analysis and Applications · 1988
The consistency and solutions of the matrix equations $A\bar X - XB = C$, $A\bar X \pm XA^T = C$, $A\bar X \pm XA^* = C$ are characterized. As a consequence it is shown that $A^T $ (respectively, $A^* $) may be obtained from A by a consimilarity transformation using a Hermitian (respectively, symmetric) matrix.