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Extremal Ranks of Some Nonlinear Matrix Expressions with Applications

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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.

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Acknowledgements

The authors would like to thank Professor Ilio Galligani and Yongge Tian and Maolin Liang and the anonymous referees for their very detailed comments and constructive suggestions, which greatly improved the presentation of this paper. This work was supported by the NSFC (Grant No. 11301397), the NSFC Mathematics TianYuan Youth Fund (Grant No. 11226126), the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (Grant No. 2012LYM-0126), and the Basic Theory and Scientific Research of Science and Technology Project of Jiangmen City, China, 2013.

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  1. School of Mathematics and Computational Science, Wuyi University, Jiangmen, 529020, Guangdong, P.R. China

    Zhiping Xiong & Yingying Qin

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  1. Zhiping Xiong
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  2. Yingying Qin
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Correspondence to Zhiping Xiong.

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Communicated by Ilio Galligani.

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Xiong, Z., Qin, Y. Extremal Ranks of Some Nonlinear Matrix Expressions with Applications. J Optim Theory Appl 163, 595–613 (2014). https://doi.org/10.1007/s10957-013-0508-0

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