Explicit determinantal formulas for solutions to the generalized Sylvester quaternion matrix equation and its special cases

TL;DR

This paper derives explicit determinantal formulas for solutions to the quaternion generalized Sylvester matrix equation and related equations, extending Cramer's rule to quaternion matrices using determinant representations.

Contribution

It introduces new determinantal representation formulas for quaternion matrix equations, including special cases and Lyapunov equations, based on quaternion determinant theory.

Findings

Explicit formulas for quaternion Sylvester equations

Determinantal representations for special cases and Lyapunov equations

Extension of Cramer's rule to quaternion matrices

Abstract

Within the framework of the theory of quaternion column-row determinants and using determinantal representations of the Moore-Penrose inverse previously obtained by the author, we get explicit determinantal representation formulas of solutions (analogs of Cramer's rule) to the quaternion two-sided generalized Sylvester matrix equation $ {\bf A}_{1}{\bf X}_{1}{\bf B}_{1}+ {\bf A}_{2}{\bf X}_{2}{\bf B}_{2}={\bf C}$ and its all special cases when its first term or both terms are one-sided. Finally, we derive determinantal representations of two like-Lyapunov equations.