Explicit Solutions of the Singular Yang--Baxter-like Matrix Equation and Their Numerical Computation

We derive several explicit formulae for finding infinitely many solutions of the equation $AXA=XAX$, when $A$ is singular. We start by splitting the equation into a couple of linear matrix equations and then show how the projectors commuting with $A$ can be used to get families containing an infinite number of solutions. Some techniques for determining those projectors are proposed, which use, in particular, the properties of the Drazin inverse, spectral projectors, the matrix sign function, and eigenvalues. We also investigate in detail how the well-known similarity transformations like Jordan and Schur decompositions can be used to obtain new representations of the solutions. The computation of solutions by the suggested methods using finite precision arithmetic is also a concern. Difficulties arising in their implementation are identified and ideas to overcome them are discussed. Numerical experiments shed some light on the methods that may be promising for solving numerically the said matrix equation.