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  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

show all references

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

[1]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076

[2]

Mingchao Zhao, You-Wei Wen, Michael Ng, Hongwei Li. A nonlocal low rank model for poisson noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021003

[3]

Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021001

[4]

Ningyu Sha, Lei Shi, Ming Yan. Fast algorithms for robust principal component analysis with an upper bound on the rank. Inverse Problems & Imaging, 2021, 15 (1) : 109-128. doi: 10.3934/ipi.2020067

[5]

Ryuji Kajikiya. Existence of nodal solutions for the sublinear Moore-Nehari differential equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1483-1506. doi: 10.3934/dcds.2020326

[6]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266

[7]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002

[8]

Peter Frolkovič, Karol Mikula, Jooyoung Hahn, Dirk Martin, Branislav Basara. Flux balanced approximation with least-squares gradient for diffusion equation on polyhedral mesh. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 865-879. doi: 10.3934/dcdss.2020350

[9]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

[10]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367

[11]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005

[12]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435

[13]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[14]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[15]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[16]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[17]

Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012

[18]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[19]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345

[20]

Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346

 Impact Factor: 

Metrics

  • PDF downloads (114)
  • HTML views (453)
  • Cited by (0)

Other articles
by authors

[Back to Top]