Xiong, Zhiping Upper and lower bounds for ranks of the matrix expression \(X-XAX\). (English) Zbl 1291.15011 Abstr. Appl. Anal. 2013, Article ID 267035, 6 p. (2013). Summary: We consider the question of how to take \(X\) such that the nonlinear matrix expression \(X-XAX\) attains its maximal and minimal possible ranks. MSC: 15A09 Theory of matrix inversion and generalized inverses 90C20 Quadratic programming PDFBibTeX XMLCite \textit{Z. Xiong}, Abstr. Appl. Anal. 2013, Article ID 267035, 6 p. (2013; Zbl 1291.15011) Full Text: DOI References: [1] Penrose, R., A genaralized inverse for matrices, Proceedings of the Cambridge Philosophical Society, 52, 406-413 (1955) · Zbl 0065.24603 [2] Ben-Iserael, A.; Greville, T. N. E., Generalized Inverses: Theorey and Applications (1974), New York, NY, USA: Wiley-Interscience, New York, NY, USA [3] Ben-Iserael, A.; Greville, T. N. E., Generalized Inverses: Theorey and Applications (2002), New York, NY, USA: Springer, New York, NY, USA [4] Wang, G.; Wei, Y.; Qiao, S., Generalized Inverses: Theory and Computations (2004), Beijing, China: Science Press, Beijing, China [5] David, C. L., Linear Algebra and Its Applications (1994), Reading, Mass, USA: Addison-Wesley, Reading, Mass, USA [6] Braden, H. N., The equations \(A^T X \pm X^T A = B\), SIAM Journal on Matrix Analysis and Applications, 20, 2, 295-302 (1999) [7] Cohan, N.; Johnson, C. R.; Rodman, L.; Woerdeman, H. J., Ranks of completions of partial matrices, Operator Theory, 40, 165-185 (1989) [8] Davis, C., Completing a matrix so as to minimize the rank, Operator Theory, 29, 87-95 (1988) [9] Gross, J., Nonnegative-definite and positive-definite solutions to the matrix equation \(A X A^* = B\)-revisited, Linear Algebra and Its Applications, 321, 1-3, 123-129 (2000) · Zbl 0984.15011 [10] Gould, N. I. M.; Hribar, M. E.; Nocedal, J., On the solution of equality constrained quadratic programming problems arising in optimization, SIAM Journal on Scientific Computing, 23, 4, 1376-1395 (2002) · Zbl 0999.65050 [11] Johnson, C. R., Matrix completion problems: a survey. In matrix theory and applications, Proceedings of Symposia in Applied Mathematics, 40, 171-197 (1990) [12] Johnson, C. R.; Whitney, G. T., Minimum rank completions, Linear and Multilinear Algebra, 28, 271-273 (1991) · Zbl 0775.15001 [13] Tian, Y., The maximal and minimal ranks of some expression of generalized inverses of matrices, Southeast Asian Bulletin of Mathematics, 25, 745-755 (2002) · Zbl 1007.15005 [14] Tian, Y.; Liu, Y., Extremal ranks of some symmetric matrix expressions with applications, SIAM Journal on Matrix Analysis and Applications, 28, 3, 890-905 (2006) · Zbl 1123.15001 [15] Tian, Y., Solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method, Nonlinear Analysis, Theory, Methods and Applications, 75, 2, 717-734 (2012) · Zbl 1236.65070 [16] Woerdeman, H. J., Minimal rank completions for block matrices, Linear Algebra and Its Applications C, 121, 105-122 (1989) · Zbl 0681.15002 [17] Xiong, Z.; Qin, Y., On the inverse of a special Schur complement, Applied Mathematics and Computation, 218, 14, 7679-7684 (2012) · Zbl 1246.65053 [18] Zhan, X.; Wang, Q. W.; Liu, X., Inertias and ranks of some Hermitian matrix functions with application, Central European Journal of Mathematics, 10, 1, 329-351 (2012) · Zbl 1253.15050 [19] Tian, Y., Upper and lower bounds for ranks of matrix expressions using generalized inverses, Linear Algebra and Its Applications, 355, 1-3, 187-214 (2002) · Zbl 1016.15003 [20] Tian, Y.; Cheng, S., The maximal and minimal ranks of \(A - B X C\) with applications, New York Journal of Mathematics, 9, 345-362 (2003) · Zbl 1036.15004 [21] Mihályffy, L., An alternative representation of the generalized inverse of partitioned matrices, Linear Algebra and Its Applications, 4, 1, 95-100 (1971) · Zbl 0236.15007 [22] Marsaglia, G.; Styan, G. P. H., Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra, 2, 3, 269-292 (1974) · Zbl 0297.15003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.