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A new expression of the Hermitian solutions to a system of matrix equations with applications. (English) Zbl 1347.15022

In the first part of this research article, the authors give a new necessary and sufficient condition for the existence of a Hermitian solution which was given earlier by C. G. Khatri and S. K. Mitra [SIAM J. Appl. Math. 31, 579–585 (1976; Zbl 0359.65033)], M. L. Arias and M. C. Gonzalez [Linear Algebra Appl. 433, No. 6, 1194–1202 (2010; Zbl 1200.47023)], Q. Wang and C. Yang [Commentat. Math. Univ. Carol. 39, No. 1, 7–13 (1998; Zbl 0937.15008)], C. R. Johnson and M. Lundquist [in: Operator theory and complex analysis. Proceedings of a workshop, held in Sapporo, Japan, June 11-14, 1991. Basel: Birkhäuser. 234–251 (1992; Zbl 0806.47010)], Q.-W. Wang and C.-Z. Dong [Linear Algebra Appl. 433, No. 7, 1481–1489 (2010; Zbl 1214.47015)], giving a suitable lemma. In the second part, the authors give a new expression of the general Hermitian solution to the system of matrix equations by giving some suitable theorems. In Parts 3 and 4, the authors give max-min ranks and inertias part of the general solution of this system with suitable theorems. This paper is written in very well manner.

MSC:

15A24 Matrix equations and identities
15A03 Vector spaces, linear dependence, rank, lineability
15A09 Theory of matrix inversion and generalized inverses
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