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# Communications in Mathematical Sciences

## Volume 17 (2019)

### Number 2

### Solving the Yang–Baxter-like matrix equation with non-diagonalizable elementary matrices

Pages: 393 – 411

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n2.a5

#### Authors

#### Abstract

Let $A=I-uv^T$, where $u$ and $v$ are two $n$-dimensional complex vectors with $v^T u=0$. Thus $A$ is not diagonalizable. We find all solutions of the quadratic matrix equation $AXA=XAX$. This is a continuation of the work [*Computers Math. Appl.*, 72(6):1541–1548, 2016] from the case of diagonalizable elementary matrices to non-diagonalizable ones.

#### Keywords

nonlinear matrix equation, elementary matrix, spectral perturbation

#### 2010 Mathematics Subject Classification

15A18

This work was supported by the National Natural Science Foundation of China (Nos. 11861008, 11501126, 11471122, 11661007, 11661008), the Research Fund of Gannan Normal University (Nos. YJG-2018-11, 18zb04), and the Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice.

Received 28 January 2018

Received revised 6 December 2018

Accepted 6 December 2018

Published 8 July 2019