### 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 |

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### Keywords

- Matrix equation
- Non-trivial solution
- Polynomial equation

### Cite this

*Electronic Journal of Linear Algebra*,

*9*, 282-289.

}

*Electronic Journal of Linear Algebra*, vol. 9, pp. 282-289.

**Non-trivial solutions to certain matrix equations.** / Li, Aihua; Randall, Duane.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Non-trivial solutions to certain matrix equations

AU - Li, Aihua

AU - Randall, Duane

PY - 2002/10/1

Y1 - 2002/10/1

N2 - 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.

AB - 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.

KW - Matrix equation

KW - Non-trivial solution

KW - Polynomial equation

UR - http://www.scopus.com/inward/record.url?scp=3042596838&partnerID=8YFLogxK

M3 - Article

VL - 9

SP - 282

EP - 289

JO - Electronic Journal of Linear Algebra

JF - Electronic Journal of Linear Algebra

SN - 1081-3810

ER -