Solving the Yang-Baxter-Like Matrix Equation With Non-Diagonalizable Elementary Matrices

Duanmei Zhou, Gannan Normal University
Guoliang Chen, East China Normal University
Jiu Ding, University of Southern Mississippi
Haiyan Tian, University of Southern Mississippi

Abstract

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