Determinantal Representations of General and (Skew-)Hermitian Solutions to the Generalized Sylvester-Type Quaternion Matrix Equation

Abstract

In this paper, we derive explicit determinantal representation formulas of general, Hermitian, and skew-Hermitian solutions to the generalized Sylvester matrix equation involving $*$-Hermicity $\mathbf{A}\mathbf{X}{\mathbf{A}}^{*}+\mathbf{B}\mathbf{Y}{\mathbf{B}}^{*}=\mathbf{C}$ over the quaternion skew field within the framework of the theory of noncommutative column-row determinants.