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.
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This research was supported by the National Natural Science Foundation of China (11226067), the Fundamental Research Funds for the Central Universities (WM1214063), China Postdoctoral Science Foundation (2012M511014).
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Yu, Sw., Song, Gj. On Solvability of Hermitian Solutions to a System of Five Matrix Equations. Mediterr. J. Math. 11, 237–253 (2014). https://doi.org/10.1007/s00009-013-0316-7
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DOI: https://doi.org/10.1007/s00009-013-0316-7