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March  2021, 11(1): 75-86. doi: 10.3934/naco.2020016

## Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C

 Faculty of Exact Sciences and Sciences of Nature and Life, Department of Mathematics and informatics, University of Oum El Bouaghi, 04000, Algeria

* Corresponding author: Sihem Guerarra

Received  June 2019 Revised  September 2019 Published  March 2021 Early access  February 2020

In this paper we derive the extremal ranks and inertias of the matrix $X+X^{\ast}-P$, with respect to $X$, where $P\in\mathbb{C} _{H}^{n\times n}$ is given, $X$ is a least rank solution to the matrix equation $AXB = C$, and then give necessary and sufficient conditions for $X+X^{\ast}\succ P$ $\left( \geq P\text{, }\prec P\text{, }\leq P\right)$ in the Löwner partial ordering. As consequence, we establish necessary and sufficient conditions for the matrix equation $AXB = C$ to have a Hermitian Re-positive or Re-negative definite solution.

Citation: Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016
##### References:
 [1] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, 2$^{\rm nd}$ ed., Springer, 2003.  Google Scholar [2] S. L. Cambell and C. D. Meyer, Generalized Inverse of Linear Transformations, SIAM, 2008. doi: 10.1137/1.9780898719048.ch0.  Google Scholar [3] J. Groβ, Nonnegative-definite and positive definite solutions to the matrix equation $AXA^{\ast} = B$-revisited, Linear Algebra Appl., 321 (2000), 123-129.  doi: 10.1016/S0024-3795(00)00033-1.  Google Scholar [4] S. Guerarra and S. Guedjiba, Common least-rank solution of matrix equations $A_{1}X_{1}B_{1} = C_{1}$ and $A_{2}X_{2} B_{2} = C_{2}$ with applications, Facta Universitatis (Niš). Ser. Math. Inform., 29 (2014), 313–323.  Google Scholar [5] S. Guerarra and S. Guedjiba, Common Hermitian least-rank solution of matrix equations $A_{1}XA_{1}^{\ast} = B_{1}$ and $A_{2}XA_{2}^{\ast} = B_{2}$ subject to inequality restrictions, Facta Universitatis (Niš). Ser. Math. Inform., 30 (2015), 539–554.  Google Scholar [6] S. Guerarra, Positive and negative definite submatrices in an Hermitian least rank solution of the matrix equation, Numer. Algebra, Contr. & Optim., 9 (2019), 15-22.   Google Scholar [7] C. G. Khatri and S. K. Mitra, Hermitian and nonnegative definite solutions of linear matrix equations, SIAM J. Appl. Math., 31 (1976), 579-585.  doi: 10.1137/0131050.  Google Scholar [8] Y. Liu, Ranks of least squares solutions of the matrix equation $AXB = C$, Comput. Mathe. Applications, 55 (2008), 1270-1278.  doi: 10.1016/j.camwa.2007.06.023.  Google Scholar [9] R. Penrose, A generalized inverse for matrices, Proc. Camb. Phil. Soc., 52 (1955), 406-413.   Google Scholar [10] P. S. Stanimirović, G-inverses and canonical forms, Facta Universitatis (Niš). Ser. Math. Inform., 15 (2000), 1–14.  Google Scholar [11] Y. Tian, Rank Equalities Related to Generalized Inverses of Matrices and Their Applications, Master Thesis, Montreal, Quebec, Canada, 2000. Google Scholar [12] Y. Tian, The maximal and minimal ranks of some expressions of generalized inverses of matrices, Southeast Asian Bull. Math., 25 (2002), 745-755.  doi: 10.1007/s100120200015.  Google Scholar [13] Y. Tian and S. Cheng, The maximal and minimal ranks of $A-BXC$ with applications, New York J. Math., 9 (2003), 345-362.   Google Scholar [14] Y. Tian, Equalities and inequalities for inertias of Hermitian matrices with applications, Linear Algebra Appl., 433 (2010), 263-296.  doi: 10.1016/j.laa.2010.02.018.  Google Scholar [15] Y. Tian, Maximization and minimization of the rank and inertias of the Hermitian matrix expression $A-BX-\left(BX\right) ^{\ast}$ with applications, Linear Algebra Appl., 434 (2011), 2109-2139.  doi: 10.1016/j.laa.2010.12.010.  Google Scholar [16] Y. Tian and H. Wang, Relations between least squares and least rank solution of the matrix equations $AXB=C$, Appl. Math. Comput., 219 (2013), 10293-10301.  doi: 10.1016/j.amc.2013.03.137.  Google Scholar [17] X. Zhang, Hermitian nonnegative-definite and positive-definite solutions of the matrix equation $AXB=C$, Appl. Math. E-Notes, 4 (2004), 40-47.   Google Scholar

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##### References:
 [1] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, 2$^{\rm nd}$ ed., Springer, 2003.  Google Scholar [2] S. L. Cambell and C. D. Meyer, Generalized Inverse of Linear Transformations, SIAM, 2008. doi: 10.1137/1.9780898719048.ch0.  Google Scholar [3] J. Groβ, Nonnegative-definite and positive definite solutions to the matrix equation $AXA^{\ast} = B$-revisited, Linear Algebra Appl., 321 (2000), 123-129.  doi: 10.1016/S0024-3795(00)00033-1.  Google Scholar [4] S. Guerarra and S. Guedjiba, Common least-rank solution of matrix equations $A_{1}X_{1}B_{1} = C_{1}$ and $A_{2}X_{2} B_{2} = C_{2}$ with applications, Facta Universitatis (Niš). Ser. Math. Inform., 29 (2014), 313–323.  Google Scholar [5] S. Guerarra and S. Guedjiba, Common Hermitian least-rank solution of matrix equations $A_{1}XA_{1}^{\ast} = B_{1}$ and $A_{2}XA_{2}^{\ast} = B_{2}$ subject to inequality restrictions, Facta Universitatis (Niš). Ser. Math. Inform., 30 (2015), 539–554.  Google Scholar [6] S. Guerarra, Positive and negative definite submatrices in an Hermitian least rank solution of the matrix equation, Numer. Algebra, Contr. & Optim., 9 (2019), 15-22.   Google Scholar [7] C. G. Khatri and S. K. Mitra, Hermitian and nonnegative definite solutions of linear matrix equations, SIAM J. Appl. Math., 31 (1976), 579-585.  doi: 10.1137/0131050.  Google Scholar [8] Y. Liu, Ranks of least squares solutions of the matrix equation $AXB = C$, Comput. Mathe. Applications, 55 (2008), 1270-1278.  doi: 10.1016/j.camwa.2007.06.023.  Google Scholar [9] R. Penrose, A generalized inverse for matrices, Proc. Camb. Phil. Soc., 52 (1955), 406-413.   Google Scholar [10] P. S. Stanimirović, G-inverses and canonical forms, Facta Universitatis (Niš). Ser. Math. Inform., 15 (2000), 1–14.  Google Scholar [11] Y. Tian, Rank Equalities Related to Generalized Inverses of Matrices and Their Applications, Master Thesis, Montreal, Quebec, Canada, 2000. Google Scholar [12] Y. Tian, The maximal and minimal ranks of some expressions of generalized inverses of matrices, Southeast Asian Bull. Math., 25 (2002), 745-755.  doi: 10.1007/s100120200015.  Google Scholar [13] Y. Tian and S. Cheng, The maximal and minimal ranks of $A-BXC$ with applications, New York J. Math., 9 (2003), 345-362.   Google Scholar [14] Y. Tian, Equalities and inequalities for inertias of Hermitian matrices with applications, Linear Algebra Appl., 433 (2010), 263-296.  doi: 10.1016/j.laa.2010.02.018.  Google Scholar [15] Y. Tian, Maximization and minimization of the rank and inertias of the Hermitian matrix expression $A-BX-\left(BX\right) ^{\ast}$ with applications, Linear Algebra Appl., 434 (2011), 2109-2139.  doi: 10.1016/j.laa.2010.12.010.  Google Scholar [16] Y. Tian and H. Wang, Relations between least squares and least rank solution of the matrix equations $AXB=C$, Appl. Math. Comput., 219 (2013), 10293-10301.  doi: 10.1016/j.amc.2013.03.137.  Google Scholar [17] X. Zhang, Hermitian nonnegative-definite and positive-definite solutions of the matrix equation $AXB=C$, Appl. Math. E-Notes, 4 (2004), 40-47.   Google Scholar
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