A System of Coupled Quaternion Matrix Equations with Seven Unknowns and Its Applications

In this paper, a system of coupled quaternion matrix equations with seven unknowns AiXi+YiBi+CiZiDi+FiWGi=Ei\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{i}X_{i}+Y_{i}B_{i}+C_{i}Z_{i}D_{i}+F_{i}WG_{i}=E_{i} \end{aligned}$$\end{document}is considered, where Ai,Bi,Ci,Di,Fi,Gi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{i},B_{i},C_{i},D_{i},F_{i},G_{i}$$\end{document} and Ei\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{i}$$\end{document} are given matrices, Xi,Yi,Zi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{i},Y_{i},Z_{i}$$\end{document} and W are unknowns (i=1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(i=1,2)$$\end{document}. Some practical necessary and sufficient conditions for the existence of a solution to this system in terms of ranks and Moore–Penrose inverses are provided. The general solution to the system is given when the solvability conditions are satisfied. Applications that are discussed include the solvability conditions and general solutions to some quaternion matrix equations involving ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document}-Hermicity. Some examples are given to illustrate the main results.