Non-Trivial Solutions to Certain Matrix Equations

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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 x n)-matrix X = (xij ) of variables together with (n x 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.

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