Some structures of submatrices in solution to the paire of matrix equations $ AX = C $, $ XB = D $

Radja Belkhiri; Sihem Guerarra

doi:10.3934/mfc.2022023

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2023, Volume 6, Issue 2: 231-252. Doi: 10.3934/mfc.2022023

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Some structures of submatrices in solution to the paire of matrix equations $ AX = C $, $ XB = D $

Radja Belkhiri and
Sihem Guerarra^,

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
^*Corresponding author: Sihem Guerarra

Received: February 2022

Revised: June 2022

Early access: July 12, 2022

Published online: July 12, 2022

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Abstract

The optimization problems involving local unitary and local contraction matrices and some Hermitian structures have been concedered in this paper. We establish a set of explicit formulas for calculating the maximal and minimal values of the ranks and inertias of the matrices $ X_{1}X_{1}^{\ast}-P_{1} $, $ X_{2}X_{2}^{\ast}-P_{1} $, $ X_{3}X_{3}^{\ast}-P_{2} $ and $ X_{4}X_{4}^{\ast }-P_{2} $, with respect to $ X_{1} $, $ X_{2} $, $ X_{3} $ and $ X_{4} $ respectively, where $ P_{1}\in \mathbb{C} ^{n_{1}\times n_{1}} $, $ P_{2}\in \mathbb{C} ^{n_{2}\times n_{2}} $ are given, $ X_{1} $, $ X_{2} $, $ X_{3} $ and $ X_{4} $ are submatrices in a general common solution $ X $ to the paire of matrix equations $ AX = C $, $ XB = D. $

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Mathematics Subject Classification: Primary: 15A24; Secondary: 15A09, 15A03.

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References

[1]	A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Corrected reprint of the 1974 original. Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980.
[2]	K. E. Chu, Singular value and generalized singular value decompositions and the solution of linear matrix equations, Linear Algebra Appl., 88/89 (1987), 83-98. doi: 10.1016/0024-3795(87)90104-2.
[3]	X. Liu, The common $(P, Q)$ generalized reflexive and anti-reflexive solutions to $AX = B$ and $XC = D$, Calcolo., 53 (2016), 227-234. doi: 10.1007/s10092-015-0146-z.
[4]	X. Liu and H. Yang, An expression of the general common least-squares solution to a pair of matrix equations with applications, Comput. Math. Appl., 61 (2011), 3071-3078. doi: 10.1016/j.camwa.2011.03.096.
[5]	X. Liu and Y. Yuan, Generalized reflexive and anti-reflexive solutions of $AX = B$, Calcolo., 53 (2016), 59-66. doi: 10.1007/s10092-014-0136-6.
[6]	G. Marsaglia and G. P. H. Styan, Equalities and inequalities for ranks of matrices, Linear. Multil. Algebra, 2 (1974/75), 269-292. doi: 10.1080/03081087408817070.
[7]	S. K. Mitra, The matrix equation $AX = C$, $XB = D$, Linear Algebra Appl., 59 (1984), 171-181. doi: 10.1016/0024-3795(84)90166-6.
[8]	R. Penrose, A generalized inverse for matrices, Proc. Camb. Phil. Soc., 51 (1955), 406-413.
[9]	Y. Qiu, Z. Zhang and J. Lu, The matrix equations $AX = B $, $CX = D$ with $PX = sXP$ constraint, Appl. Math. Comput., 189 (2007), 1428-1434. doi: 10.1016/j.amc.2006.12.046.
[10]	P. S. Stanimirovic, G-inverses and canonical forms, Facta Univ. Ser. Math. Inform., 15 (2000), 1-14.
[11]	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.
[12]	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.
[13]	Q. Wang, The general solution to a system of real quaternion matrix equations, Comput. Math. Appl., 49 (2005), 665-675. doi: 10.1016/j.camwa.2004.12.002.
[14]	Q. Wang, X. Zhang and Z. He, On the Hermitian structures of the solution to a pair of matrix equations, Lineair. Multil. Algebra., 61 (2013), 73-90. doi: 10.1080/03081087.2012.663370.
[15]	Y. Yuan, Least-squares solutions to the matrix equations $AX = B$, $XC = D$, Appl. Math. Comput., 216 (2010), 3120-3125. doi: 10.1016/j.amc.2010.04.002.
[16]	F. Zhang, Matrix Theory: Basic Results and Techniques, 2$^{nd}$ edition, Springer, New York, 2011. doi: 10.1007/978-1-4614-1099-7.
[17]	Q. Zhang and Q. W. Wang, The $(P, Q)$-(skew) symmetric extremal rank solutions to a system of quaternion matrix equations, Appl. Math. Comput., 217 (2011), 9286-9296. doi: 10.1016/j.amc.2011.04.011.