A magic determinant formula for symmetric polynomials of eigenvalues

2020/09/03 by Jules Jacobs, Jacobs, Jules
Physics and Astronomy · #15-02 #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2009.01345

openalex publication_date 2020/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Symmetric polynomials of the roots of a polynomial can be written as polynomials of the coefficients, and by applying this to the characteristic polynomial we can write a symmetric polynomial of the eigenvalues a_i of an n× n matrix A as a polynomial of the entries of the matrix. We give a magic formula for this: symbolically substitute a↦ A in the symmetric polynomial and replace multiplication by det. For instance, for a 2×2 matrix A with eigenvalues a₁,a₂, a₁ a₂² +a₁² a₂ amp; =det(A₁, A₂²)+ det(A₁², A₂) where A_i^k is the i-th column of A^k. One may also take negative powers, allowing us to calculate: a₁a₂^-1+a₁^-1a₂ amp; =det(A₁,A₂^-1)+det(A₁^-1,A₂) The magic method also works for multivariate symmetric polynomials of the eigenvalues of a set of commuting matrices, e.g. for 2×2 matrices A and B with eigenvalues a₁,a₂ and b₁,b₂, a₁ b₁ a₂² + a₁²a₂b₂ amp; = det(AB₁,A₂²) + det(A₁²,AB₂)

Citations

• Algebra II

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