Zhang, Xiang; Xiang, Shu-Wen Solving optimization problems on Hermitian matrix functions with applications. (English) Zbl 1268.15016 J. Appl. Math. 2013, Article ID 593549, 11 p. (2013). Summary: We consider the extremal inertias and ranks of the matrix expressions \(f(X, Y) = A_3 - B_3X - (B_3X)^\ast - C_3YD_3 - (C_3YD_3)^\ast\), where \(A_3 = A_3^\ast\), \(B_3\), \(C_3\), and \(D_3\) are known matrices and \(Y\) and \(X\) are the solutions to the matrix equations \(A_1Y = C_1\), \(YB_1 = D_1\), and \(A_2X = C_2\), respectively. As applications, we present necessary and sufficient conditions for the previous matrix function \(f(X, Y)\) to be positive (negative), nonnegative (positive) definite or nonsingular. We also characterize the relations between the Hermitian part of the solutions of the above-mentioned matrix equations. Furthermore, we establish necessary and sufficient conditions for the solvability of the system of matrix equations \(A_1Y = C_1\), \(YB_1 = D_1\), \(A_2X = C_2\), and \(B_3X + (B_3X)^\ast + C_3YD_3 + (C_3YD_3)^\ast = A_3\), and give an expression of the general solution to the above-mentioned system when it is solvable. MSC: 15A24 Matrix equations and identities 15A18 Eigenvalues, singular values, and eigenvectors 15A03 Vector spaces, linear dependence, rank, lineability Keywords:Hermitian matrix functions; extremal inertias; rank; system of matrix equations PDFBibTeX XMLCite \textit{X. Zhang} and \textit{S.-W. Xiang}, J. Appl. Math. 2013, Article ID 593549, 11 p. (2013; Zbl 1268.15016) Full Text: DOI References: [1] H. Zhang, “Rank equalities for Moore-Penrose inverse and Drazin inverse over quaternion,” Annals of Functional Analysis, vol. 3, no. 2, pp. 115-127, 2012. · Zbl 1255.15006 [2] L. Bao, Y. Lin, and Y. 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