Basis-Free Analysis of Singular Tuples and Eigenpairs of Tensors

TL;DR

This paper introduces an intrinsic, basis-free framework for defining and analyzing eigenvalues, eigenvectors, singular values, and singular vectors of tensors, connecting tensor analysis with pure mathematics.

Contribution

It provides the first basis-free definitions of tensor eigenvalues and singular values, simplifying conceptual understanding and linking tensor analysis to algebraic geometry and differential topology.

Findings

Basis-free definitions clarify tensor concepts.

Morse theory applied to symmetric tensors.

Reproof of classical results in a new framework.

Abstract

A tensor in applied mathematics is usually defined as a multidimensional array of numbers. This presumes a choice of basis in $\mathbb{R}^n$ or in some other vector space, and tensorial concepts are defined accordingly. In this article we define eigenvalues, eigenvectors, singular values, and singular vectors of a tensor intrinsically, without reference to a basis. The basis-free approach has several advantages. First, it shows more clearly the relationship between tensor analysis and areas of pure mathematics such as abstract algebra, differential topology, and algebraic geometry. Second, it obviates the need to prove that a concept defined in terms of coordinates is independent of the choice of basis. Third, an intrinsic definition is usually conceptually simpler. As illustrations we show how Morse theory from differential topology can be used to analyze eigenvalues and eigenvectors of a symmetric tensor. We also reprove a few results that are obvious in the basis-free approach, but not otherwise.