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

Assaf Goldberger;  Neumann, Michael — 2004

Czechoslovak Mathematical Journal

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 ( λ I - 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 ( λ I - A ) + ∑ i = 1 n - 1 ρ n - 2 i | a i | ( λ - ρ ) i ≤ λ n - ρ n , for all λ ≥ ρ . We use this inequality to derive the inequality that: ∏ 2 n ( ρ - λ i ) ≤ ρ n - 2 ∑ i = 2 n ( ρ - λ i ) . In the spirit of a celebrated conjecture due to Boyle-Handelman,...

Bounds on the subdominant eigenvalue involving group inverses with applications to graphs

Stephen J. Kirkland;  Neumann, Michael;  Bryan L. Shader — 1998

Czechoslovak Mathematical Journal

Let A be an n × n symmetric, irreducible, and nonnegative matrix whose eigenvalues are λ 1 > λ 2 ≥ ... ≥ λ n . In this paper we derive several lower and upper bounds, in particular on λ 2 and λ n , but also, indirectly, on μ = max 2 ≤ i ≤ n | λ i | . The bounds are in terms of the diagonal entries of the group generalized inverse, Q # , of the singular and irreducible M-matrix Q = λ 1 I - A . Our starting point is a spectral resolution for Q # . We consider the case of equality in some of these inequalities and we apply our results to the algebraic connectivity of undirected...

On a bound on algebraic connectivity: the case of equality

Stephen J. Kirkland;  Neumann, Michael;  Bryan L. Shader — 1998

Czechoslovak Mathematical Journal

In a recent paper the authors proposed a lower bound on 1 - λ i , where λ i , λ i ≠ 1 , is an eigenvalue of a transition matrix T of an ergodic Markov chain. The bound, which involved the group inverse of I - T , was derived from a more general bound, due to Bauer, Deutsch, and Stoer, on the eigenvalues of a stochastic matrix other than its constant row sum. Here we adapt the bound to give a lower bound on the algebraic connectivity of an undirected graph, but principally consider the case of equality in the bound when...

A central limit theorem for triangular arrays of weakly dependent random variables, with applications in statistics

Michael H. Neumann — 2013

ESAIM: Probability and Statistics

We derive a central limit theorem for triangular arrays of possibly nonstationary random variables satisfying a condition of weak dependence in the sense of Doukhan and Louhichi [84 (1999) 313–342]. The proof uses a new variant of the Lindeberg method: the behavior of the partial sums is compared to that of partial sums of Gaussian random variables. We also discuss a few applications in statistics which show that our central limit theorem is tailor-made for statistics of different type.

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