Communications in Mathematical Sciences
Volume 17 (2019)
Solving the Yang–Baxter-like matrix equation with non-diagonalizable elementary matrices
Pages: 393 – 411
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.
nonlinear matrix equation, elementary matrix, spectral perturbation
2010 Mathematics Subject Classification
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