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Bounds for the Z-eigenpair of general nonnegative tensors

Qilong Liu, Yaotang Li (2016)

Open Mathematics

In this paper, we consider the Z-eigenpair of a tensor. A lower bound and an upper bound for the Z-spectral radius of a weakly symmetric nonnegative irreducible tensor are presented. Furthermore, upper bounds of Z-spectral radius of nonnegative tensors and general tensors are given. The proposed bounds improve some existing ones. Numerical examples are reported to show the effectiveness of the proposed bounds.

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...

Brownian motion tree models are toric

Bernd Sturmfels, Caroline Uhler, Piotr Zwiernik (2020)

Kybernetika

Felsenstein's classical model for Gaussian distributions on a phylogenetic tree is shown to be a toric variety in the space of concentration matrices. We present an exact semialgebraic characterization of this model, and we demonstrate how the toric structure leads to exact methods for maximum likelihood estimation. Our results also give new insights into the geometry of ultrametric matrices.

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