Non-trivial solutions to certain matrix equations

The existence of non-trivial solutions X to matrix equations of the form F (X,A1,A2, · · · ,As) = G(X,A1,A2, · · · ,As) over the real numbers is investigated. Here F and G denote monomials in the (n × n)-matrix X = (xij) of variables together with (n × n)-matrices A1,A2, · · · ,As for s ≥ 1 and n ≥ 2 such that F and G have different total positive degrees in X. An example with s = 1 is given by F (X,A) = X2AX and G(X,A) = AXA where deg(F ) = 3 and deg(G) = 1. The Borsuk-Ulam Theorem guarantees that a non-zero matrix X exists satisfying the matrix equation F (X,A1,A2, · · · ,As) = G(X,A1,A2, · · · ,As) in (n2 − 1) components whenever F and G have different total odd degrees in X. The Lefschetz Fixed Point Theorem guarantees the existence of special orthogonal matrices X satisfying matrix equations F (X,A1,A2, · · · ,As) = G(X,A1,A2, · · · ,As) whenever deg(F ) > deg(G) ≥ 1, A1,A2, · · · ,As are in SO(n), and n ≥ 2. Explicit solution matrices X for the equations with s = 1 are constructed. Finally, nonsingular matrices A are presented for which X2AX = AXA admits no non-trivial solutions.