For a given matrix $ G \in \mathbb{R}^{l \times n} $, define a subspace $ \Omega $ by $ \Omega = \left\{ z \in \mathbb{R}^n \mid Gz = 0 \right\} $, and let $ \mathcal{H} = \left\{ K \in \mathbb{R}^{n \times n} \mid (x, Ky) = (Kx, y), \, x^\top Kx \ge 0, \, \forall x, y \in \Omega \right\} $. Thus, $ \mathcal{H} $ consists of all symmetric nonnegative definite matrices on the subspace $ \Omega $. In this paper, we investigate the existence of $ X\in \mathcal{H} $ satisfying the linear matrix equations $ AX = C, XB = D $, and $ AXB = D $. A necessary and sufficient condition for solvability and an explicit representation of the general solution are obtained. Finally, two numerical examples are provided to demonstrate the accuracy of the results.
Citation: Yinlan Chen, Qingyi Yang, Honglin Zou. Symmetric nonnegative definite solutions of certain linear matrix equations on a subspace[J]. AIMS Mathematics, 2026, 11(7): 21727-21742. doi: 10.3934/math.2026879
For a given matrix $ G \in \mathbb{R}^{l \times n} $, define a subspace $ \Omega $ by $ \Omega = \left\{ z \in \mathbb{R}^n \mid Gz = 0 \right\} $, and let $ \mathcal{H} = \left\{ K \in \mathbb{R}^{n \times n} \mid (x, Ky) = (Kx, y), \, x^\top Kx \ge 0, \, \forall x, y \in \Omega \right\} $. Thus, $ \mathcal{H} $ consists of all symmetric nonnegative definite matrices on the subspace $ \Omega $. In this paper, we investigate the existence of $ X\in \mathcal{H} $ satisfying the linear matrix equations $ AX = C, XB = D $, and $ AXB = D $. A necessary and sufficient condition for solvability and an explicit representation of the general solution are obtained. Finally, two numerical examples are provided to demonstrate the accuracy of the results.
| [1] |
S. K. Mitra, Common solutions to a pair of linear matrix equations $A_1XB_1 = C_1$ and $A_2XB_2 = C_2$, Math. Proc. Cambridge, 74 (1973), 213–216. https://doi.org/10.1017/S030500410004799X doi: 10.1017/S030500410004799X
|
| [2] |
Q. Wang, Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations, Comput. Math. Appl., 49 (2005), 641–650. https://doi.org/10.1016/j.camwa.2005.01.014 doi: 10.1016/j.camwa.2005.01.014
|
| [3] |
Y. Yuan, Least-squares solutions to the matrix equations $AX = B$ and $XC = D$, Appl. Math. Comput., 216 (2010), 3120–3125. https://doi.org/10.1016/j.amc.2010.04.002 doi: 10.1016/j.amc.2010.04.002
|
| [4] |
A. Daji$\acute{c}$, J. J. Koliha, Positive solutions to the equations $AX = C$ and $XB = D$ for Hilbert space operators, J. Math. Anal. Appl., 333 (2007), 567–576. https://doi.org/10.1016/j.jmaa.2006.11.016 doi: 10.1016/j.jmaa.2006.11.016
|
| [5] |
R. Penrose, A generalized inverse for matrices, Math. Proc. Cambridge, 51 (1955), 406–413. https://doi.org/10.1017/S0305004100030401 doi: 10.1017/S0305004100030401
|
| [6] |
H. Dai, On the symmetric solutions of linear matrix equations, Linear Algebra Appl., 131 (1990), 1–7. https://doi.org/10.1016/0024-3795(90)90370-R doi: 10.1016/0024-3795(90)90370-R
|
| [7] |
C. G. Khatri, S. K. Mitra, Hermitian and nonnegative definite solutions of linear matrix equations, SIAM J. Appl. Math., 31 (1976), 579–585. https://doi.org/10.1137/0131050 doi: 10.1137/0131050
|
| [8] |
Y. Peng, X. Hu, L. Zhang, An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation $AXB = C$, Appl. Math. Comput., 160 (2005), 763–777. https://doi.org/10.1016/j.amc.2003.11.030 doi: 10.1016/j.amc.2003.11.030
|
| [9] |
D. S. Cvetkovi$\acute{c}$, Re-nnd solutions of the matrix equation $AXB = C$, J. Aust. Math. Soc., 84 (2008), 63–72. https://doi.org/10.1017/s1446788708000207 doi: 10.1017/s1446788708000207
|
| [10] |
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. https://doi.org/10.1016/j.amc.2015.01.098 doi: 10.1016/j.amc.2015.01.098
|
| [11] |
D. Zhang, T. Jiang, C. Jiang, G. Wang, A complex structure-preserving algorithm for computing the singular value decomposition of a quaternion matrix and its applications, Numer. Algor., 95 (2024), 267–283. https://doi.org/10.1007/s11075-023-01571-4 doi: 10.1007/s11075-023-01571-4
|
| [12] |
D. Zhang, T. Jiang, G. Wang, V. I. Vasil'ev, Two efficient algorithms for the commutative quaternion equality constrained least squares problem, J. Comput. Appl. Math., 474 (2026), 116964. https://doi.org/10.1016/j.cam.2025.116964 doi: 10.1016/j.cam.2025.116964
|
| [13] | Y. Yuan, H. Dai, A class of inverse problem for matrices on subspace (Chinese), Numer. Math.-J. Chin. Univ., 27 (2005), 69–77. |
| [14] |
S. Hu, Y. Yuan, Common solutions to the matrix equations $AX = B$ and $XC = D$ on a subspace, J. Optim. Theory Appl., 198 (2023), 372–386. https://doi.org/10.1007/s10957-023-02247-8 doi: 10.1007/s10957-023-02247-8
|
| [15] |
S. Hu, Y. Yuan, The anti-symmetric solution of the matrix equation $AXA^\top = B$ on a null subspace, Results Math., 79 (2024), 100. https://doi.org/10.1007/s00025-024-02129-z doi: 10.1007/s00025-024-02129-z
|
| [16] |
M. Zeng, Y. Yuan, Sparse structure preserving finite element model updating problem for gyroscopic systems, Mech. Syst. Signal Proc., 203 (2023), 110734. https://doi.org/10.1016/j.ymssp.2023.110734 doi: 10.1016/j.ymssp.2023.110734
|
| [17] |
M. Mohammadi, P. Moradi, M. Jalili, SCE: subspace-based core expansion method for community detection in complex networks, Physica A, 527 (2019), 121084. https://doi.org/10.1016/j.physa.2019.121084 doi: 10.1016/j.physa.2019.121084
|
| [18] | K. Berahmand, F. Saberi-Movahed, R. Sheikhpour, Y. Li, M. Jalili, A comprehensive survey on spectral clustering with graph structure learning, arXiv: 2501.13597. https://doi.org/10.48550/arXiv.2501.13597 |
| [19] |
F. Saberi-Movahed, K. Berahmand, R. Sheikhpour, Y. Li, S. Pan, M. Jalili, Nonnegative matrix factorization in dimensionality reduction: a survey, ACM Comput. Surv., 58 (2025), 118. https://doi.org/10.1145/3767726 doi: 10.1145/3767726
|
| [20] | A. Ben-Israel, T. Greville, Generalized inverses: theory and applications, New York: Springer, 2003. https://doi.org/10.1007/b97366 |
| [21] |
L. Zhang, The solvability conditions for the inverse problem of symmetric nonnegative definite matrices (Chinese), Mathematica Numerica Sinica, 11 (1989), 337–343. https://doi.org/10.12286/jssx.1989.4.337 doi: 10.12286/jssx.1989.4.337
|
| [22] | H. Dai, The stability of solutions for two classes of inverse problems of matrices (Chinese), Numer. Math.-J. Chin. Univ., 1 (1994), 87–96. |
| [23] |
P. D. Young, D. M Young, M. M. Young, A general Hermitian nonnegative-definite solution to the matrix equation $AXB = C$, Advances in Linear Algebra and Matrix Theory, 7 (2017), 7–17. https://doi.org/10.4236/alamt.2017.71002 doi: 10.4236/alamt.2017.71002
|
Yinlan Chen, Qingyi Yang, Honglin Zou. Symmetric nonnegative definite solutions of certain linear matrix equations on a subspace[J]. AIMS Mathematics, 2026, 11(7): 21727-21742. doi: 10.3934/math.2026879
DownLoad: