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 $q\mapsto \hat q$

TL;DR

This paper extends Roth's solvability criteria for matrix equations over quaternions with involution, providing necessary and sufficient conditions for solutions to exist in this non-commutative setting.

Contribution

It generalizes Roth's criteria to quaternionic matrix equations involving an involutive automorphism, broadening the understanding of their solvability conditions.

Findings

Criteria established for equations $AX-oxed{ ext{hat}} XB=C$ and $X-Aoxed{ ext{hat}} XB=C$

Conditions involve similarity of block matrices over quaternions

Extends classical results to non-commutative quaternionic setting

Abstract

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 a field by W.E. Roth (1952) and over the skew field of quaternions by Huang Liping (1996). H.K. Wimmer (1988) obtained an analogous criterion for the matrix equation $X-AXB=C$ over a field. We extend these criteria to the matrix equations $AX-\widehat XB=C$ and $X-A\widehat XB=C$ over the skew field of quaternions with a fixed involutive automorphism $q\mapsto \hat q$.