The Dominant Eigenvalue of an Essentially Nonnegative Tensor

TL;DR

This paper extends the concept of essentially nonnegative matrices to higher order tensors, proving convexity properties of their dominant eigenvalues and providing an algorithm for their computation with numerical validation.

Contribution

It introduces the notion of essentially nonnegative tensors, establishes convexity of their dominant eigenvalues, and develops an effective algorithm for calculating these eigenvalues.

Findings

Spectral radius is the dominant eigenvalue for nonnegative tensors.

Convexity and log convexity properties are proven for these eigenvalues.

An algorithm effectively computes the dominant eigenvalue with supporting numerical results.

Abstract

It is well known that the dominant eigenvalue of a real essentially nonnegative matrix is a convex function of its diagonal entries. This convexity is of practical importance in population biology, graph theory, demography, analytic hierarchy process and so on. In this paper, the concept of essentially nonnegativity is extended from matrices to higher order tensors, and the convexity and log convexity of dominant eigenvalues for such a class of tensors are established. Particularly, for any nonnegative tensor, the spectral radius turns out to be the dominant eigenvalue and hence possesses these convexities. Finally, an algorithm is given to calculate the dominant eigenvalue, and numerical results are reported to show the effectiveness of the proposed algorithm.