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Rank/inertia approaches to weighted least-squares solutions of linear matrix equations

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  • Jiang, Bo
  • Tian, Yongge

Abstract

The well-known linear matrix equation AX=B is the simplest representative of all linear matrix equations. In this paper, we study quadratic properties of weighted least-squares solutions of this matrix equation. We first establish two groups of closed-form formulas for calculating the global maximum and minimum ranks and inertias of matrices in the two quadratical matrix-valued functions Q1−XP1X′ and Q2−X′P2X subject to the restriction trace[(AX−B)′W(AX−B)]=min, where both Pi and Qi are real symmetric matrices, i=1,2,W is a positive semi-definite matrix, and X′ is the transpose of X. We then use the rank and inertia formulas to characterize quadratic properties of weighted least-squares solutions of AX=B, including necessary and sufficient conditions for weighted least-squares solutions of AX=B to satisfy the quadratic symmetric matrix equalities XP1X′=Q1 an X′P2X=Q2, respectively, and necessary and sufficient conditions for the quadratic matrix inequalities XP1X′≻Q1 (≽Q1, ≺Q1, ≼Q1) and X′P2X≻Q2 (≽Q2, ≺Q2, ≼Q2) in the Löwner partial ordering to hold, respectively. In addition, we give closed-form solutions to four Löwner partial ordering optimization problems on Q1−XP1X′ and Q2−X′P2X subject to weighted least-squares solutions of AX=B.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:315:y:2017:i:c:p:400-413
    DOI: 10.1016/j.amc.2017.07.079
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    References listed on IDEAS

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    1. Yongge Tian & Wenxing Guo, 2016. "On comparison of dispersion matrices of estimators under a constrained linear model," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 25(4), pages 623-649, November.
    2. Dong, Baomin & Guo, Wenxing & Tian, Yongge, 2014. "On relations between BLUEs under two transformed linear models," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 279-292.
    3. Y. Tian, 2017. "Some equalities and inequalities for covariance matrices of estimators under linear model," Statistical Papers, Springer, vol. 58(2), pages 467-484, June.
    4. 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.
    5. Yongge Tian, 2015. "A new derivation of BLUPs under random-effects model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(8), pages 905-918, November.
    6. 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.
    Full references (including those not matched with items on IDEAS)

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