Roth's solvability criteria for the matrix equations AX−XˆB=C and X−AXˆB=C over the skew field of quaternions with an involutive automorphism q↦qˆ

The matrix equation AX−XB=CAX−XB=C has a solution if and only if the matrices [AC0B] and [A00B] are similar. This criterion was proved over a field by W.E. Roth (1952) and over the skew field of quaternions by Huang Liping (1996). H.K. Wimmer (1988) proved that the matrix equation X−AXB=CX−AXB=C over a field has a solution if and only if the matrices [AC0I] and [I00B] are simultaneously equivalent to [A00I] and [I00B]. We extend these criteria to the matrix equations AX−XˆB=C and X−AXˆB=C over the skew field of quaternions with a fixed involutive automorphism q↦qˆ.