Extremal Ranks of Some Nonlinear Matrix Expressions with Applications

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.

[1]  B. Zheng,et al.  The least squares g-inverses for sum of matrices , 2013 .

[2]  Yongge Tian,et al.  A simultaneous decomposition of a matrix triplet with applications , 2011, Numer. Linear Algebra Appl..

[3]  Yongge Tian.  Upper and lower bounds for ranks of matrix expressions using generalized inverses , 2002 .

[4]  Yongge Tian.  Maximization and minimization of the rank and inertia of the Hermitian matrix expression A — BX — (BX)* with applications , 2011 .

[5]  R. Penrose.  A Generalized inverse for matrices , 1955 .

[6]  Yongge Tian.  Solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method , 2012 .

[7]  J. Crouzeix,et al.  Definiteness and semidefiniteness of quadratic forms revisited , 1984 .

[8]  Yongge Tian,et al.  Extremal Ranks of Some Symmetric Matrix Expressions with Applications , 2006, SIAM J. Matrix Anal. Appl..

[9]  Yongge Tian,et al.  The maximal and minimal ranks of A − BXC with applications , 2003 .

[10]  Musheng Wei,et al.  On rank-constrained Hermitian nonnegative-definite least squares solutions to the matrix equation AXA H =B , 2007 .

[11]  G. Styan,et al.  THREE RANK FORMULAS ASSOCIATED WITH THE COVARIANCE MATRICES OF THE BLUE AND THE OLSE IN THE GENERAL LINEAR MODEL , 2005, Econometric Theory.

[12]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

[13]  Sujit Kumar Mitra,et al.  Hermitian and Nonnegative Definite Solutions of Linear Matrix Equations , 1976 .

[14]  Yongge Tian.  Completing triangular block matrices with maximal and minimal ranks , 2000 .

[15]  Range invariance of certain matrix products , 1983 .

[16]  H. W. Braden,et al.  The Equations ATX\pm XTA=B , 1999, SIAM J. Matrix Anal. Appl..

[17]  Chandler Davis.  Completing a Matrix so as to Minimize the Rank , 1988 .

[18]  Yongge Tian,et al.  The Minimum Rank of a 3 × 3 Partial Block Matrix , 2002 .

[19]  Yoshio Takane,et al.  Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA∗=B , 2009 .

[20]  Katta G. Murty,et al.  Some NP-complete problems in quadratic and nonlinear programming , 1987, Math. Program..

[21]  A. Berman,et al.  Improvement of a Large Analytical Model Using Test Data , 1983 .

[22]  Vincent D. Blondel,et al.  Proceedings of the 2000 American Control Conference , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[23]  Bing Zheng,et al.  The reverse order laws for {1, 2, 3} - and {1, 2, 4} -inverses of a two-matrix product , 2008, Appl. Math. Lett..

[24]  Yong Hui Liu,et al.  Ranks of Solutions of the Linear Matrix Equation AX + YB = C , 2006, Comput. Math. Appl..

[25]  J. Geelen.  Maximum rank matrix completion , 1999 .

[26]  Yongge Tian,et al.  More on extremal ranks of the matrix expressions A − BX ± X*B* with statistical applications , 2008, Numer. Linear Algebra Appl..

[27]  Zhiping Xiong,et al.  Invariance properties of an operator product involving generalized inverses , 2011 .

[28]  G. Styan,et al.  Equalities and Inequalities for Ranks of Matrices , 1974 .

[29]  Stephen P. Boyd,et al.  Rank minimization and applications in system theory , 2004, Proceedings of the 2004 American Control Conference.

[30]  Douglas P. Wiens,et al.  On equality and proportionality of ordinary least squares, weighted least squares and best linear unbiased estimators in the general linear model , 2006 .

[31]  Yongge Tian.  Formulas for calculating the extremum ranks and inertias of a four-term quadratic matrix-valued function and their applications , 2012 .