Block tensors and symmetric embeddings

TL;DR

This paper explores how to embed general tensors into symmetric tensors to relate their spectral properties and ranks, extending matrix concepts to higher-order tensors.

Contribution

It introduces a method for embedding order-d tensors into symmetric tensors, linking their spectral properties and ranks.

Findings

Embedding tensors preserves spectral properties

Power methods for tensors relate to symmetric embeddings

Tensor rank doubles under symmetric embedding

Abstract

Well known connections exist between the singular value decomposition of a matrix A and the Schur decomposition of its symmetric embedding sym(A) = [ 0 A; A' 0]. In particular, if s is a singular value of A then +s and -s are eigenvalues of the symmetric embedding. The top and bottom halves of sym(A)'s eigenvectors are singular vectors for A. Power methods applied to A can be related to power methods applied to sym(A). The rank of sym(A) is twice the rank of A. In this paper we show how to embed a general order-d tensor A into an order-d symmetric tensor sym(A). Through the embedding we relate (a) power methods for A's singular values to power methods for sym(A)'s eigenvalues and (b) the rank of A to the rank of sym(A).