Cramer’s Rules for Sylvester Quaternion Matrix Equation and Its Special Cases

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 A1X1B1+A2X2B2=C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbf{A}_{1}{} \mathbf{X}_{1}{} \mathbf{B}_{1}+ \mathbf{A}_{2}{} \mathbf{X}_{2}{} \mathbf{B}_{2}=\mathbf{C}$$\end{document} and its all special cases when its first term or both terms are one-sided. Finally, determinantal representations of solutions to like-Lyapunov equations are derived.