Commuting Solutions of a Quadratic Matrix Equation for Nilpotent Matrices
Qixiang Dong, Jiu Ding, Qianglian Huang · Algebra Colloquium · 2018
We solve the quadratic matrix equation AXA = XAX with a given nilpotent matrix A, to find all commuting solutions. We first provide a key lemma, and consider the special case that A has only one Jordan block to motivate the idea for the general case. Our main result gives the structure of all the commuting solutions of the equation with an arbitrary nilpotent matrix.