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