# American Institute of Mathematical Sciences

March  2019, 9(1): 15-22. doi: 10.3934/naco.2019002

## Positive and negative definite submatrices in an Hermitian least rank solution of the matrix equation AXA*=B

 Faculty of exact sciences and sciences of nature and life, Department of Mathematics, University of Oum El Bouaghi, 04000, Algeria

* Corresponding author: Sihem Guerarra

Received  September 2017 Revised  April 2018 Published  October 2018

This work is devoted to establish the extremal inertias ofthe two submatrices $X_{1}$ and $X_{4}$ in a Hermitian least rank solution $X$of the matrix equation $AXA^{*}=B$. From these formulas, necessary andsufficient conditions for these submatrices to be positive (nonpositive,negative, nonnegative) definite are achieved.

Citation: Sihem Guerarra. Positive and negative definite submatrices in an Hermitian least rank solution of the matrix equation AXA*=B. Numerical Algebra, Control and Optimization, 2019, 9 (1) : 15-22. doi: 10.3934/naco.2019002
##### References:
 [1] A. Ben Israel and T. Greville, Generalized Inverse, Theory and Applications, 2nd edition, Springer, New York, 2003. doi: 10.1007/978-1-4612-0873-0. [2] S. L. Cambell and C. D. Meyer, Generalized Inverse of Linear Transformations, Society for Industrial and Applied Mathematics, 2008. doi: 10.1007/978-1-4612-0873-0. [3] S. Guerarra and S. Guedjiba, Common Hermitian least-rank solution of matrix equations A1XA1* = B1 and A2XA2* = B2 subject to inequality restrictions, Facta universitatis (Niš). Ser. Math. Inform., 30 (2015), 539-554.  doi: 10.2307/2152750. [4] Y. Liu and Y. Tian, Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA* = B with applications, J. Appl. Math. Comput., 32 (2010), 289-301.  doi: 10.2307/2152750. [5] Y. Liu, Y. Tian and Y. Takane, Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA* = B, Linear Algebra Appl., 431 (2009), 2359-2372.  doi: 10.2307/2152750. [6] G. Marsaglia and G.P.H. Styan, Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra., 2 (1974), 269-292.  doi: 10.2307/2152750. [7] Y. Tian, The maximal and minimal ranks of some expressions of generalized inverses of matrices, Southeast Asian Bull. Math., 25 (2002), 745-755.  doi: 10.2307/2152750. [8] Y. Tian, Equalities and inequalities for inertias of Hermitian matrices with applications, Linear Algebra Appl., 433 (2010), 263-296.  doi: 10.2307/2152750. [9] Y. Tian, Maximization and minimization of the rank and inertias of the Hermitian matrix expression A - BX - (BX)* with applications, Linear Algebra Appl., 434 (2011), 2109-2139.  doi: 10.2307/2152750. [10] Y. Tian, Least-squares solutions and least-rank solutions of the matrix equation AXA* = B and their relations, Numer. Linear Algebra Appl., 20 (2013), 713-722.  doi: 10.2307/2152750. [11] Y. Tian and S. Cheng, The maximal and minimal ranks of A - BXC with applications, New York Journal of Mathematics, 9 (2003), 345-362.  doi: 10.2307/2152750.

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##### References:
 [1] A. Ben Israel and T. Greville, Generalized Inverse, Theory and Applications, 2nd edition, Springer, New York, 2003. doi: 10.1007/978-1-4612-0873-0. [2] S. L. Cambell and C. D. Meyer, Generalized Inverse of Linear Transformations, Society for Industrial and Applied Mathematics, 2008. doi: 10.1007/978-1-4612-0873-0. [3] S. Guerarra and S. Guedjiba, Common Hermitian least-rank solution of matrix equations A1XA1* = B1 and A2XA2* = B2 subject to inequality restrictions, Facta universitatis (Niš). Ser. Math. Inform., 30 (2015), 539-554.  doi: 10.2307/2152750. [4] Y. Liu and Y. Tian, Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA* = B with applications, J. Appl. Math. Comput., 32 (2010), 289-301.  doi: 10.2307/2152750. [5] Y. Liu, Y. Tian and Y. Takane, Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA* = B, Linear Algebra Appl., 431 (2009), 2359-2372.  doi: 10.2307/2152750. [6] G. Marsaglia and G.P.H. Styan, Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra., 2 (1974), 269-292.  doi: 10.2307/2152750. [7] Y. Tian, The maximal and minimal ranks of some expressions of generalized inverses of matrices, Southeast Asian Bull. Math., 25 (2002), 745-755.  doi: 10.2307/2152750. [8] Y. Tian, Equalities and inequalities for inertias of Hermitian matrices with applications, Linear Algebra Appl., 433 (2010), 263-296.  doi: 10.2307/2152750. [9] Y. Tian, Maximization and minimization of the rank and inertias of the Hermitian matrix expression A - BX - (BX)* with applications, Linear Algebra Appl., 434 (2011), 2109-2139.  doi: 10.2307/2152750. [10] Y. Tian, Least-squares solutions and least-rank solutions of the matrix equation AXA* = B and their relations, Numer. Linear Algebra Appl., 20 (2013), 713-722.  doi: 10.2307/2152750. [11] Y. Tian and S. Cheng, The maximal and minimal ranks of A - BXC with applications, New York Journal of Mathematics, 9 (2003), 345-362.  doi: 10.2307/2152750.
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