We derive the necessary and sufficient conditions of and the expressions for the orthogonal solutions, the symmetric orthogonal solutions, and the skew-symmetric orthogonal solutions of the system of matrix equations $AX=B$ and $XC=D$, respectively. When the matrix equations are not consistent, the least squares symmetric orthogonal solutions and the least squares skew-symmetric orthogonal solutions are respectively given. As an auxiliary, an algorithm is provided to compute the least squares symmetric orthogonal solutions, and meanwhile an example is presented to show that it is reasonable.

Let $R$ be an $n$ by $n$ nontrivial real symmetric involution matrix, that is,$R=R^{-1}=R^T\ne I_n$. An $n\times n$ complex matrix $A$ is termed $R$-conjugate if$\overline{A}=RAR$, where $\overline{A}$ denotes the conjugate of $A$. We give necessary and sufficientconditions for the existence of the Hermitian $R$-conjugate solution to the systemof complex matrix equations $AX=C\hspace{0.17em}\text{and}\hspace{0.17em}XB=D$ and present an expression ofthe Hermitian $R$-conjugate solution to this system when the solvability conditionsare satisfied. In addition, the solution to an optimal approximation problem isobtained. Furthermore, the least squares Hermitian $R$-conjugate solution with theleast norm to this system mentioned above is considered. The representation ofsuch solution is also derived. Finally, an algorithm and numerical examples aregiven.

We consider the perturbation analysis of the matrix equation $X-{\sum }_{i=1}^m{A}_i^{\ast }X{A}_i+{\sum }_{j=1}^n{B}_j^{\ast }X{B}_j=I$. Based on the matrix differentiation, we first give a precise perturbation boundfor the positive definite solution. A numerical example is presented to illustrate the sharpnessof the perturbation bound.

Some new Kantorovich-type inequalities for Hermitian matrix are proposed in this paper. We consider what happens to these inequalities when the positive definite matrix is allowed to be invertible and provides refinements of the classical results.