## Abstract

The existence of non-trivial solutions X to matrix equations of the form F(X,A_{1},A_{2}, ⋯ ,A_{s}) = G(X,A_{1},A_{2}, ⋯ ,A_{s}) over the real numbers is investigated. Here F and G denote monomials in the (n x n)-matrix X = (x_{ij} ) of variables together with (n x n)-matrices A_{1},A_{2}, ⋯ ,A_{s} 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) = X^{2}AX 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,A_{1},A_{2}, ⋯ ,A_{s}) = G(X,A_{1},A_{2}, ⋯ ,A_{s}) in (n^{2} - 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,A_{1},A_{2}, ⋯ ,A_{s}) = G(X,A_{1},A_{2}, ⋯ ,A_{s}) whenever deg(F) > deg(G) ≥ 1, A_{1},A_{2}, ⋯ ,A_{s} 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 X^{2}AX = AXA admits no non-trivial solutions.

Original language | English |
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Pages (from-to) | 282-289 |

Number of pages | 8 |

Journal | Electronic Journal of Linear Algebra |

Volume | 9 |

State | Published - 1 Oct 2002 |

## Keywords

- Matrix equation
- Non-trivial solution
- Polynomial equation