Explicit solutions of the Yang–Baxter-like matrix equation for a diagonalizable matrix with spectrum contained in {1, α, 0}

Abstract

Let A ∈ Cl × l be a diagonalizable matrix with spectrum contained in the set {1, α, 0}. In this paper we derive a general and explicit expression for the solutions X of the Yang–Baxter-like matrix equation AXA=XAX. The idea involves partitioning the matrices to discuss four block-matrix equations and utilizing the eigen-properties of the matrices obtained. When A is an idempotent matrix, the result generates the formula obtained in Mansour et al. (2017). We give examples to illustrate the validity of the results obtained in this note.