An improvement of an inequality of fiedler leading to a new conjecture on nonnegative matrices

Assaf Goldberger, Michael Neumann

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Abstract

Suppose that A is an n × n nonnegative matrix whose eigenvalues are λ = ρ(A), λ 2,..., λ n. Fiedler and others have shown that det(λI - A) ≤ λ n - ρ n, for all λ > ρ, with equality for any such λ if and only if A is the simple cycle matrix. Let a i be the signed sum of the determinants of the principal submatrices of A of order i × i, i=1,..., n - 1. We use similar techniques to Fiedler to show that Fiedler's inequality can be strengthened to: det(λI - A) + ∑ i=1 n-1 ρ n-2i |a i| (λ - ρ) i ≤ λ n - ρ n, for all λ ≥ ρ. We use this inequality to derive the inequality that: ∏ 2 k(ρ - λ i) ≤ ρ n-2i=2 n (ρ - λ i). In the spirit of a celebrated conjecture due to Boyle-Handelman, this inequality inspires us to conjecture the following inequality on the nonzero eigenvalues of A: If λ 1 = ρ(A), λ 2,..., λ k are (all) the nonzero eigenvalues of A, then ∏ 2 k(ρ - λ i) ≤ ρ k-2i=2 k (ρ - λ). We prove this conjecture for the case when the spectrum of A is real.

Original languageEnglish
Pages (from-to)773-780
Number of pages8
JournalCzechoslovak Mathematical Journal
Volume54
Issue number3
DOIs
StatePublished - Sep 2004
Externally publishedYes

Funding

FundersFunder number
National Science Foundation

    Keywords

    • M-matrices
    • determinants
    • nonnegative matrices

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