On Solvability of Hermitian Solutions to a System of Five Matrix Equations

In this paper, we establish the formulas of the maximal and minimal ranks of the Hermitian matrix expression where X (1) and X (2) are Hermitian solutions to two systems of matrix equations A (1) X (1) = C (1),X (1) B (1) = C (2) and A (2) X (2) = C (3),X (2) B (2) = C (4), respectively. Using this result and matrix rank method, we give necessary and sufficient conditions for the existence of Hermitian solutions to a system of five matrix equations by rank equalities. The general expressions and extreme ranks of the Hermitian solutions X (1) and X (2) to the system mentioned above are also presented.