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Solving Optimization Problems on Hermitian Matrix Functions with Applications

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  • Xiang Zhang
  • Shu-Wen Xiang

Abstract

We consider the extremal inertias and ranks of the matrix expressions , where , and are known matrices and and are the solutions to the matrix equations , , and , respectively. As applications, we present necessary and sufficient condition for the previous matrix function 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 , , , and , and give an expression of the general solution to the above-mentioned system when it is solvable.

Suggested Citation

  • Xiang Zhang & Shu-Wen Xiang, 2013. "Solving Optimization Problems on Hermitian Matrix Functions with Applications," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-11, April.
    • Handle: RePEc:hin:jnljam:593549
    DOI: 10.1155/2013/593549
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