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Extremal Ranks of Some Nonlinear Matrix Expressions with Applications

Author

Listed:
  • Zhiping Xiong

    (Wuyi University)

  • Yingying Qin

    (Wuyi University)

Abstract

In this paper, we establish a group of closed-form formulas for the maximal and minimal ranks of a nonlinear matrix expression with respect to two variant matrices by using a linearization method and some known formulas for extremal ranks of linear matrix expressions. In addition, by using some pure algebraic operations of matrices and their generalized inverses, we derive the maximal and minimal ranks of the above nonlinear matrix expression, where the two variant matrices are any solutions of two consistent matrix equations. As an application, we derive some sufficient and necessary conditions for the existence of the solution of a nonlinear matrix function.

Suggested Citation

  • Zhiping Xiong & Yingying Qin, 2014. "Extremal Ranks of Some Nonlinear Matrix Expressions with Applications," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 595-613, November.
    • Handle: RePEc:spr:joptap:v:163:y:2014:i:2:d:10.1007_s10957-013-0508-0
    DOI: 10.1007/s10957-013-0508-0
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    References listed on IDEAS

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    1. Tian, Yongge & Wiens, Douglas P., 2006. "On equality and proportionality of ordinary least squares, weighted least squares and best linear unbiased estimators in the general linear model," Statistics & Probability Letters, Elsevier, vol. 76(12), pages 1265-1272, July.
    2. Puntanen, Simo & Styan, George P.H. & Tian, Yongge, 2005. "Three Rank Formulas Associated With The Covariance Matrices Of The Blue And The Olse In The General Linear Model," Econometric Theory, Cambridge University Press, vol. 21(3), pages 659-663, June.
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