Roth's solvability criteria for the matrix equations AX - (X)over-capB = C and X - A(X)over-capB = C over the skew field of quaternions with an involutive automorphism q -> (q)over-cap

摘要

The matrix equation AX - XB = C has a solution if and only if the matrices [A C 0 B] and [A 0 o B] 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 = C over a field has a solution if and only if the matrices [A C 0 I] and [I 0 L B] are simultaneously equivalent to [A 0 0 I] and [I 0 0 B]. We extend these criteria to the matrix equations AX - XB = C and X - A (X) over capB = C over the skew field of quaternions with a fixed involutive automorphism q -> (q) over cap. (C) 2016 Elsevier Inc. All rights reserved.