A Spectral Theory for Tensors

TL;DR

This paper introduces a comprehensive spectral theory for tensors, including a novel factorization that generalizes eigenvalues, eigenvectors, and orthogonality from matrices to tensors, enabling hierarchical spectral analysis.

Contribution

It develops a unified multilinear algebra framework for tensor spectral theory, including tensor decomposition, eigen-objects, and characteristic polynomials, extending matrix concepts to tensors.

Findings

Tensor can be decomposed into orthogonal and scaling tensors.

The framework generalizes matrix hermicity and transpose to tensors.

A recursive spectral hierarchy for tensors is induced.

Abstract

In this paper we propose a general spectral theory for tensors. Our proposed factorization decomposes a tensor into a product of orthogonal and scaling tensors. At the same time, our factorization yields an expansion of a tensor as a summation of outer products of lower order tensors . Our proposed factorization shows the relationship between the eigen-objects and the generalised characteristic polynomials. Our framework is based on a consistent multilinear algebra which explains how to generalise the notion of matrix hermicity, matrix transpose, and most importantly the notion of orthogonality. Our proposed factorization for a tensor in terms of lower order tensors can be recursively applied so as to naturally induces a spectral hierarchy for tensors.