A unified treatment for the restricted solutions of the matrix equation $AXB=C$

Jiao Xu; Hairui Zhang; Lina Liu; Huiting Zhang; Yongxin Yuan; Jiao Xu; Hairui Zhang; Lina Liu; Huiting Zhang; Yongxin Yuan

doi:10.3934/math.2020424

AIMS Mathematics

2020, Volume 5, Issue 6: 6594-6608. doi: 10.3934/math.2020424

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Research article

A unified treatment for the restricted solutions of the matrix equation $AXB=C$

School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, China

Received: 18 June 2020 Accepted: 18 August 2020 Published: 24 August 2020
MSC : 15A09, 15A24

In this paper, the Hermitian, skew-Hermitian, Re-nonnegative definite, Re-positive definite, Re-nonnegative definite least-rank and Re-positive definite least-rank solutions of the matrix equation $AXB = C$ are considered. The necessary and sufficient condition for the existence of such type of solution to the equation is provided and the explicit expression of the general solution is also given.
Citation: Jiao Xu, Hairui Zhang, Lina Liu, Huiting Zhang, Yongxin Yuan. A unified treatment for the restricted solutions of the matrix equation $AXB=C$[J]. AIMS Mathematics, 2020, 5(6): 6594-6608. doi: 10.3934/math.2020424

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Abstract

In this paper, the Hermitian, skew-Hermitian, Re-nonnegative definite, Re-positive definite, Re-nonnegative definite least-rank and Re-positive definite least-rank solutions of the matrix equation $AXB = C$ are considered. The necessary and sufficient condition for the existence of such type of solution to the equation is provided and the explicit expression of the general solution is also given.

References

[1]	Y. Chen, Z. Peng, T. Zhou, LSQR iterative common symmetric solutions to matrix equations AXB = E and CXD = F, Appl. Math. Comput., 217 (2010), 230-236.
[2]	M. K. Zak, F. Toutounian, Nested splitting conjugate gradient method for matrix equation AXB = C and preconditioning, Comput. Math. Appl., 66 (2013), 269-278. doi: 10.1016/j.camwa.2013.05.004
[3]	X. Wang, Y. Li, L. Dai, On Hermitian and skew-Hermitian splitting iteration methods for the linear matrix equation AXB = C, Comput. Math. Appl., 65 (2013), 657-664. doi: 10.1016/j.camwa.2012.11.010
[4]	Z. Tian, M. Tian, Z. Liu, et al., The Jacobi and Gauss-Seidel-type iteration methods for the matrix equation AXB = C, Appl. Math. Comput., 292 (2017), 63-75.
[5]	Z. Liu, Z. Li, C. Ferreira, et al., Stationary splitting iterative methods for the matrix equation AXB = C, Appl. Math. Comput., 378 (2020), 125195.
[6]	Y. Yuan, H. Dai, Generalized reflexive solutions of the matrix equation AXB = D and an associated optimal approximation problem, Comput. Math. Appl., 56 (2008), 1643-1649. doi: 10.1016/j.camwa.2008.03.015
[7]	F. Zhang, M. Wei, Y. Li, et al., The minimal norm least squares Hermitian solution of the complex matrix equation AXB + CXD = E, J. Franklin I., 355 (2018), 1296-1310. doi: 10.1016/j.jfranklin.2017.12.023
[8]	G. Song, S. Yu, Nonnegative definite and Re-nonnegative definite solutions to a system of matrix equations with statistical applications, Appl. Math. Comput., 338 (2018), 828-841.
[9]	H. Liu, Y. Yuan, An inverse problem for symmetric matrices in structural dynamic model updating, Chinese J. Eng. Math., 26 (2009), 1083-1089.
[10]	M. I. Friswell, J. E. Mottershead, Finite element model updating in structural dynamics, Kluwer Academic Publishers: Dordrecht, 1995.
[11]	F. Tisseur, K. Meerbergen, The quadratic eigenvalue problem, SIAM Review, 43 (2001), 235-286. doi: 10.1137/S0036144500381988
[12]	V. L. Mehrmann, The autonomous linear quadratic control problem: theory and numerical solution, In: Lecture Notes in Control and Information Sciences, 163, Springer, Heidelberg, 1991.
[13]	G. Duan, S. Xu, W. Huang, Generalized positive definite matrix and its application in stability analysis, Acta Mechanica Sinica, 21 (1989), 754-757. (in Chinese)
[14]	G. Duan, R. J. Patton, A note on Hurwitz stability of matrices, Automatica, 34 (1998), 509-511. doi: 10.1016/S0005-1098(97)00217-3
[15]	C. G. Khatri, S. K. Mitra, Hermitian and nonnegative definite solutions of linear matrix equations, SIAM J. Appl. Math., 31 (1976), 579-585. doi: 10.1137/0131050
[16]	A. Navarra, P. L. Odell, D. M. Young, A representation of the general common solution to the matrix equations A1XB1 = C₁ and A₂XB₂ = C₂ with applications, Comput. Math. Appl., 41 (2001), 929-935. doi: 10.1016/S0898-1221(00)00330-8
[17]	F. Zhang, Y. Li, W. Guo, et al., Least squares solutions with special structure to the linear matrix equation AXB = C, Appl. Math. Comput., 217 (2011), 10049-10057.
[18]	Q. Wang, C. Yang, The Re-nonnegative definite solutions to the matrix equation AXB = C, Commentationes Mathematicae Universitatis Carolinae, 39 (1998), 7-13.
[19]	D. S. Cvetković-Ilić, Re-nnd solutions of the matrix equation AXB = C, J. Aust. Math. Soc., 84 (2008), 63-72. doi: 10.1017/S1446788708000207
[20]	X. Zhang, L. Sheng, Q. Xu, A note on the real positive solutions of the operator equation AXB = C, Journal of Shanghai Normal University (Natural Sciences), 37 (2008), 454-458.
[21]	Y. Yuan, K. Zuo, The Re-nonnegative definite and Re-positive definite solutions to the matrix equation AXB = D, Appl. Math. Comput., 256 (2015), 905-912.
[22]	L. Wu, The Re-positive definite solutions to the matrix inverse problem AX = B, Linear Algebra Appl., 174 (1992), 145-151.
[23]	L. Wu, B. Cain, The Re-nonnegative definite solutions to the matrix inverse problem AX = B, Linear Algebra Appl., 236 (1996), 137-146. doi: 10.1016/0024-3795(94)00142-1
[24]	J. Groß, Explicit solutions to the matrix inverse problem AX = B, Linear Algebra Appl., 289 (1999), 131-134. doi: 10.1016/S0024-3795(97)10008-8
[25]	X. Liu, Comments on "The common Re-nnd and Re-pd solutions to the matrix equations AX = C and XB = D", Appl. Math. Comput., 236 (2014), 663-668.
[26]	A. Ben-Israel, T. N. E. Greville, Generalized Inverses: Theory and Applications (second edition), Springer, New York, 2003.
[27]	H. W. Braden, The equations $A^\top X \pm X^\top A=B$, SIAM J. Matrix Anal. Appl., 20 (1998), 295-302.
[28]	Y. Yuan, On the symmetric solutions of a class of linear matrix equation, Chinese J. Eng. Math., 15 (1998), 25-29.
[29]	L. Mihályffy, An alternative representation of the generalized inverse of partitioned matrices, Linear Algebra Appl., 4 (1971), 95-100. doi: 10.1016/0024-3795(71)90031-0
[30]	A. Albert, Conditions for positive and nonnegative definiteness in terms of pseudoinverses, SIAM J. Appl. Math., 17 (1969), 434-440. doi: 10.1137/0117041
[31]	Y. Tian, H. Wang, Relations between least-squares and least-rank solutions of the matrix equation AXB = C, Appl. Math. Comput., 219 (2013), 10293-10301.

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