# 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 & 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.

show all references

##### 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.
 [1] Yun Cai, Song Li. Convergence and stability of iteratively reweighted least squares for low-rank matrix recovery. Inverse Problems & Imaging, 2017, 11 (4) : 643-661. doi: 10.3934/ipi.2017030 [2] Li-Fang Dai, Mao-Lin Liang, Wei-Yuan Ma. Optimization problems on the rank of the solution to left and right inverse eigenvalue problem. Journal of Industrial & Management Optimization, 2015, 11 (1) : 171-183. doi: 10.3934/jimo.2015.11.171 [3] Jaeyoung Byeon, Sungwon Cho, Junsang Park. On the location of a peak point of a least energy solution for Hénon equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1055-1081. doi: 10.3934/dcds.2011.30.1055 [4] Hubert L. Bray, Marcus A. Khuri. A Jang equation approach to the Penrose inequality. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 741-766. doi: 10.3934/dcds.2010.27.741 [5] Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic & Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159 [6] Relinde Jurrius, Ruud Pellikaan. On defining generalized rank weights. Advances in Mathematics of Communications, 2017, 11 (1) : 225-235. doi: 10.3934/amc.2017014 [7] Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401 [8] Futoshi Takahashi. On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1237-1241. doi: 10.3934/cpaa.2013.12.1237 [9] Arthur Henrique Caixeta, Irena Lasiecka, Valéria Neves Domingos Cavalcanti. On long time behavior of Moore-Gibson-Thompson equation with molecular relaxation. Evolution Equations & Control Theory, 2016, 5 (4) : 661-676. doi: 10.3934/eect.2016024 [10] Luciano Abadías, Carlos Lizama, Marina Murillo-Arcila. Hölder regularity for the Moore-Gibson-Thompson equation with infinite delay. Communications on Pure & Applied Analysis, 2018, 17 (1) : 243-265. doi: 10.3934/cpaa.2018015 [11] Marta Pellicer, Joan Solà-Morales. Optimal scalar products in the Moore-Gibson-Thompson equation. Evolution Equations & Control Theory, 2019, 8 (1) : 203-220. doi: 10.3934/eect.2019011 [12] Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31 [13] Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006 [14] Yangyang Xu, Ruru Hao, Wotao Yin, Zhixun Su. Parallel matrix factorization for low-rank tensor completion. Inverse Problems & Imaging, 2015, 9 (2) : 601-624. doi: 10.3934/ipi.2015.9.601 [15] Yongge Tian. A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 289-326. doi: 10.3934/naco.2015.5.289 [16] Xianchao Xiu, Lingchen Kong. Rank-one and sparse matrix decomposition for dynamic MRI. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 127-134. doi: 10.3934/naco.2015.5.127 [17] Irena PawŁow. The Cahn--Hilliard--de Gennes and generalized Penrose--Fife models for polymer phase separation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2711-2739. doi: 10.3934/dcds.2015.35.2711 [18] H. D. Scolnik, N. E. Echebest, M. T. Guardarucci. Extensions of incomplete oblique projections method for solving rank-deficient least-squares problems. Journal of Industrial & Management Optimization, 2009, 5 (2) : 175-191. doi: 10.3934/jimo.2009.5.175 [19] Henri Berestycki, Juncheng Wei. On least energy solutions to a semilinear elliptic equation in a strip. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1083-1099. doi: 10.3934/dcds.2010.28.1083 [20] Marko Nedeljkov, Sanja Ružičić. On the uniqueness of solution to generalized Chaplygin gas. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4439-4460. doi: 10.3934/dcds.2017190

Impact Factor: