Decomposition of tensors

TL;DR

This paper proves the existence of a unique strongly orthogonal tensor decomposition that characterizes all critical points of multilinear forms, confirming a conjecture about finiteness.

Contribution

It establishes a unique strongly orthogonal decomposition for tensors that determines all critical points, solving a conjecture on finiteness.

Findings

Existence of a unique strongly orthogonal tensor decomposition.

Characterization of all critical points of multilinear forms.

Confirmation of Friedland's finiteness conjecture.

Abstract

We consider representations of tensors as sums of decomposable tensors or, equivalently, decomposition of multilinear forms into one--forms. In this short note we show that there exists a particular finite strongly orthogonal decomposition which is essentially unique and yields all critical points of the multilinear form on the torus. In particular, this determines exactly the number of critical points of the multilinear form, giving an affirmative answer to a finiteness conjecture by Friedland.