TY - JOUR

T1 - An improvement of an inequality of fiedler leading to a new conjecture on nonnegative matrices

AU - Goldberger, Assaf

AU - Neumann, Michael

N1 - Funding Information:
Research supported in part by NSF Grant No.

PY - 2004/9

Y1 - 2004/9

N2 - 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-2 ∑ i=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-2 ∑ i=2 k (ρ - λ). We prove this conjecture for the case when the spectrum of A is real.

AB - 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-2 ∑ i=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-2 ∑ i=2 k (ρ - λ). We prove this conjecture for the case when the spectrum of A is real.

KW - M-matrices

KW - determinants

KW - nonnegative matrices

UR - http://www.scopus.com/inward/record.url?scp=11144254033&partnerID=8YFLogxK

U2 - 10.1007/s10587-004-6425-5

DO - 10.1007/s10587-004-6425-5

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AN - SCOPUS:11144254033

SN - 0011-4642

VL - 54

SP - 773

EP - 780

JO - Czechoslovak Mathematical Journal

JF - Czechoslovak Mathematical Journal

IS - 3

ER -