Block tensors and symmetric embeddings

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).