Let A = PQ^T, where P and Q are two n × 2 complex matrices of full column rank such that Q^TP is singular. We solve the quadratic matrix equation AXA = XAX. Together with a previous paper devoted to the case that Q^TP is nonsingular, we have completely solved the matrix equation with any given matrix A of rank-two.

Let A be a singular diagonalizable complex matrix with three distinct eigenvalues. We derive all explicit solutions X of the Yang-Baxter-like matrix equation AXA = XAX, by taking advantage of the Jordan form structure of A. The result generates the formula obtained in Chen et al. (2019) and M. Saeed Ibrahim Adam et al. (2019). We give examples to illustrate the validity of the results obtained in this paper.

Let A be a nilpotent matrix of index two, and consider the Yang–Baxter-like matrix equation AXA=XAX. We first obtain a system of matrix equations of smaller sizes to find all the solutions of the original matrix equation. When A is a nilpotent matrix with rank 1 and rank 2, we get all solutions of the Yang–Baxter-like matrix equation.

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