SOLUTIONS OF THE YANG-BAXTER-LIKE MATRIX EQUATION FOR THE MATRIX WITH NONSINGULAR JORDAN BLOCKS

Saijie Chen; Xiaoli Li; Lanping Zhu; Qianglian Huang

doi:10.11948/20220267

2023 Volume 13 Issue 2

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Saijie Chen, Xiaoli Li, Lanping Zhu, Qianglian Huang. SOLUTIONS OF THE YANG-BAXTER-LIKE MATRIX EQUATION FOR THE MATRIX WITH NONSINGULAR JORDAN BLOCKS[J]. Journal of Applied Analysis & Computation, 2023, 13(2): 986-999. doi: 10.11948/20220267

Citation:

Saijie Chen, Xiaoli Li, Lanping Zhu, Qianglian Huang. SOLUTIONS OF THE YANG-BAXTER-LIKE MATRIX EQUATION FOR THE MATRIX WITH NONSINGULAR JORDAN BLOCKS[J]. Journal of Applied Analysis & Computation, 2023, 13(2): 986-999. doi: 10.11948/20220267

SOLUTIONS OF THE YANG-BAXTER-LIKE MATRIX EQUATION FOR THE MATRIX WITH NONSINGULAR JORDAN BLOCKS

1.
School of Mathematical Sciences, Yangzhou University, 225002, China

Corresponding author: Email: huangql@yzu.edu.cn (Q. Huang)
Fund Project: The authors were supported by National Natural Science Foundation of China (11771378) and the Postgraduate Research and Practice Innovation Program of Jiangsu Province (KYCX21-3189)

Abstract

Abstract

Let $ A $ be a nonsingular matrix with only one Jordan block, we prove that the Yang-Baxter-like matrix equation $ AXA=XAX $ has no nonzero singular solution. When $ A $ is a nonsingular matrix with at least two Jordan blocks, the ranks of all nonzero singular solutions are obtained. This provides a necessary condition for a matrix to be a solution of the Yang-Baxter-like matrix equation. As applications, we obtain a family of nontrivial solutions for the nonsingular Jordan block with $ 3\times 3 $, and further investigate the non-commuting solutions for the nonsingular matrix with $ n\times n. $
- Yang-Baxter-like matrix equation /
- group inverse /
- non-commuting solution /
- Jordan block
MSC: 15A24, 15A09

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References

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