Some structures of submatrices in solution to the paire of matrix equations $ AX = C $, $ XB = D $

The optimization problems involving local unitary and local contraction matrices and some Hermitian structures have been concedered in this paper. We establish a set of explicit formulas for calculating the maximal and minimal values of the ranks and inertias of the matrices \begin{document}$ X_{1}X_{1}^{\ast}-P_{1} $\end{document}, \begin{document}$ X_{2}X_{2}^{\ast}-P_{1} $\end{document}, \begin{document}$ X_{3}X_{3}^{\ast}-P_{2} $\end{document} and \begin{document}$ X_{4}X_{4}^{\ast }-P_{2} $\end{document}, with respect to \begin{document}$ X_{1} $\end{document}, \begin{document}$ X_{2} $\end{document}, \begin{document}$ X_{3} $\end{document} and \begin{document}$ X_{4} $\end{document} respectively, where \begin{document}$ P_{1}\in \mathbb{C} ^{n_{1}\times n_{1}} $\end{document}, \begin{document}$ P_{2}\in \mathbb{C} ^{n_{2}\times n_{2}} $\end{document} are given, \begin{document}$ X_{1} $\end{document}, \begin{document}$ X_{2} $\end{document}, \begin{document}$ X_{3} $\end{document} and \begin{document}$ X_{4} $\end{document} are submatrices in a general common solution \begin{document}$ X $\end{document} to the paire of matrix equations \begin{document}$ AX = C $\end{document}, \begin{document}$ XB = D. $\end{document}