Mediterranean Journal of Mathematics

, Volume 11, Issue 2, pp 237–253

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

• Shao-wen Yu
• Guang-jing Song
Article

## Abstract

In this paper, we establish the formulas of the maximal and minimal ranks of the Hermitian matrix expression $${C_{5}-A_{3}X_{1}A_{3}*-A_{4}X_{2}A_{4}*}$$ 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 $${A_{1}X_{1}=C_{1},X_{1}B_{1}=C_{2},A_{2}X_{2}=C_{3},X_{2}B_{2}=C_{4} ,A_{3}X_{1}A_{3}*+A_{4}X_{2}A_{4}*=C_{5}}$$ 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.

## Mathematics Subject Classification (2000)

15A03 15A09 15A24 15A33

## Keywords

System of Matrix equations minimal rank maximal rank Hermitian solution generalized inverse

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