Solving Optimization Problems on Hermitian Matrix Functions with Applications

Xiang Zhang and Shu-Wen Xiang

Journal of Applied Mathematics, 2013, vol. 2013, issue 1

Abstract: We consider the extremal inertias and ranks of the matrix expressions f(X, Y) = A3 − B3X − (B3X) * − C3YD3 − (C3YD3) *, where A3=A3*, B3, C3, and D3 are known matrices and Y and X are the solutions to the matrix equations A1Y = C1, YB1 = D1, and A2X = C2, respectively. As applications, we present necessary and sufficient condition for the previous matrix function f(X, Y) to be positive (negative), non‐negative (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 A1Y = C1, YB1 = D1, A2X = C2, and B3X + (B3X) * + C3YD3 + (C3YD3) * = A3, and give an expression of the general solution to the above‐mentioned system when it is solvable.

Date: 2013
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https://doi.org/10.1155/2013/593549

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnljam:v:2013:y:2013:i:1:n:593549

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