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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 |
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.
References:
[1] |
A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, 2$^{\rm nd}$ ed., Springer, 2003. |
[2] |
S. L. Cambell and C. D. Meyer, Generalized Inverse of Linear Transformations, SIAM, 2008.
doi: 10.1137/1.9780898719048.ch0. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[9] |
R. Penrose,
A generalized inverse for matrices, Proc. Camb. Phil. Soc., 52 (1955), 406-413.
|
[10] |
P. S. Stanimirović, G-inverses and canonical forms, Facta Universitatis (Niš). Ser. Math. Inform., 15 (2000), 1–14. |
[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. |
[13] |
Y. Tian and S. Cheng,
The maximal and minimal ranks of $A-BXC$ with applications, New York J. Math., 9 (2003), 345-362.
|
[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. |
[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. |
[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. |
[17] |
X. Zhang,
Hermitian nonnegative-definite and positive-definite solutions of the matrix equation $AXB=C$, Appl. Math. E-Notes, 4 (2004), 40-47.
|
show all references
References:
[1] |
A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, 2$^{\rm nd}$ ed., Springer, 2003. |
[2] |
S. L. Cambell and C. D. Meyer, Generalized Inverse of Linear Transformations, SIAM, 2008.
doi: 10.1137/1.9780898719048.ch0. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[9] |
R. Penrose,
A generalized inverse for matrices, Proc. Camb. Phil. Soc., 52 (1955), 406-413.
|
[10] |
P. S. Stanimirović, G-inverses and canonical forms, Facta Universitatis (Niš). Ser. Math. Inform., 15 (2000), 1–14. |
[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. |
[13] |
Y. Tian and S. Cheng,
The maximal and minimal ranks of $A-BXC$ with applications, New York J. Math., 9 (2003), 345-362.
|
[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. |
[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. |
[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. |
[17] |
X. Zhang,
Hermitian nonnegative-definite and positive-definite solutions of the matrix equation $AXB=C$, Appl. Math. E-Notes, 4 (2004), 40-47.
|
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