Consimilarity and quaternion matrix equations $AX-\hat{X}B=C$, $X-A\hat{X}B=C$

TL;DR

This paper develops a new canonical form for quaternion matrices under a specific involutive automorphism and applies it to solve related matrix equations involving quaternion conjugation.

Contribution

It introduces an analogous canonical form for quaternion matrices under a broader class of automorphisms and applies it to analyze quaternion matrix equations.

Findings

Canonical form for quaternion matrices under involutive automorphisms

Solution methods for quaternion matrix equations involving conjugation

Extension of previous consimilarity results to more general automorphisms

Abstract

L.Huang [Linear Algebra Appl. 331 (2001) 21-30] gave a canonical form of a quaternion matrix $A$ with respect to consimilarity transformations $\tilde{S}^{-1}AS$ in which $S$ is a nonsingular quaternion matrix and $\tilde{h}:=a-bi+cj-dk$ for each quaternion $h=a+bi+cj+dk$. We give an analogous canonical form of a quaternion matrix with respect to consimilarity transformations $\hat{S}^{-1}AS$ in which $h\mapsto\hat{h}$ is an arbitrary involutive automorphism of the skew field of quaternions. We apply the obtained canonical form to the quaternion matrix equations $AX-\hat{X}B=C$ and $X-A\hat{X}B=C$.