The Dimension of Eigenvariety of Nonnegative Tensors Associated with Spectral Radius

TL;DR

This paper proves that for nonnegative weakly irreducible tensors, the set of eigenvectors associated with the spectral radius is finite, and characterizes when this set can have positive dimension, with applications to hypergraph adjacency tensors.

Contribution

It establishes the zero-dimensionality of the eigenvariety for weakly irreducible tensors and characterizes conditions for higher-dimensional eigenvarieties in nonnegative tensors.

Findings

Eigenvariety dimension is zero for weakly irreducible tensors.

Characterization of nonnegative symmetric tensors with positive eigenvariety dimension.

Application of results to hypergraph adjacency tensors.

Abstract

For a nonnegative weakly irreducible tensor, its spectral radius is an eigenvalue corresponding to a unique positive eigenvector up to a scalar called the Perron vector. But including the Perron vector, it may have more than one eigenvector corresponding to the spectral radius. The projective eigenvariety associated with the spectral radius is the set of the eigenvectors corresponding to the spectral radius considered in the complex projective space. In this paper we prove that the dimension of the above projective eigenvariety is zero, i.e. there are finite many eigenvectors associated with the spectral radius up to a scalar. For a general nonnegative tensor, we characterize the nonnegative combinatorially symmetric tensor for which the dimension of projective eigenvariety associated with spectral radius is greater than zero. Finally we apply those results to the adjacency tensors of uniform hypergraphs.