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Max-Min Problems on the Ranks and Inertias of the Matrix Expressions A−BXC±(BXC)∗ with Applications

Author

Listed:
  • Yonghui Liu

    (Shanghai Finance University)

  • Yongge Tian

    (Central University of Finance and Economics)

Abstract

We introduce a simultaneous decomposition for a matrix triplet (A,B,C ∗), where A=±A ∗ and (⋅)∗ denotes the conjugate transpose of a matrix, and use the simultaneous decomposition to solve some conjectures on the maximal and minimal values of the ranks of the matrix expressions A−BXC±(BXC)∗ with respect to a variable matrix X. In addition, we give some explicit formulas for the maximal and minimal values of the inertia of the matrix expression A−BXC−(BXC)∗ with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression D−CXC ∗ subject to Hermitian solutions of a consistent matrix equation AXA ∗=B, as well as the extremal ranks and inertias of the Hermitian Schur complement D−B ∗ A ∼ B with respect to a Hermitian generalized inverse A ∼ of A. Various consequences of these extremal ranks and inertias are also presented in the paper.

Suggested Citation

  • Yonghui Liu & Yongge Tian, 2011. "Max-Min Problems on the Ranks and Inertias of the Matrix Expressions A−BXC±(BXC)∗ with Applications," Journal of Optimization Theory and Applications, Springer, vol. 148(3), pages 593-622, March.
    • Handle: RePEc:spr:joptap:v:148:y:2011:i:3:d:10.1007_s10957-010-9760-8
    DOI: 10.1007/s10957-010-9760-8
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    Citations

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    Cited by:

    1. Song, Guangjing & Yu, Shaowen, 2018. "Nonnegative definite and Re-nonnegative definite solutions to a system of matrix equations with statistical applications," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 828-841.
    2. Yongge Tian & Bo Jiang, 2017. "Quadratic properties of least-squares solutions of linear matrix equations with statistical applications," Computational Statistics, Springer, vol. 32(4), pages 1645-1663, December.
    3. Jiang, Bo & Tian, Yongge, 2017. "Rank/inertia approaches to weighted least-squares solutions of linear matrix equations," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 400-413.
    4. Shao-Wen Yu, 2012. "The Real and Complex Hermitian Solutions to a System of Quaternion Matrix Equations with Applications," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2012, pages 1-19, January.
    5. Shanshan Hu & Yongxin Yuan, 2023. "Common Solutions to the Matrix Equations $$AX=B$$ A X = B and $$XC=D$$ X C = D on a Subspace," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 372-386, July.
    6. Liu, Xifu, 2015. "The Hermitian solution of AXA*=B subject to CXC* ≥ D," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 890-898.

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