Solving the Yang-Baxter-like matrix equation for a class of elementary matrices

Let u and v be two n-dimensional vectors in the complex field such that vTu≠0, and let A=I−uvT. We find all the solutions of the quadratic matrix equation AXA=XAX in various situations, with the help of a spectral perturbation result for rank-one updated matrices. Two numerical examples are given to demonstrate the effectiveness of the method, including an application of the result to a 4×4 equation related to the classic Yang-Baxter equation.