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)=([0A;AT0]). In particular, if σ is a singular value of A then +σ and -σ 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 develop similar connections for tensors by building on L.-H. Lim’s variational approach to tensor singular values and vectors. We show how to embed a general order-d tensor A into an order-d symmetric tensor sym(A). Through the embedding we relate power methods for A’s singular values to power methods for sym(A)’s eigenvalues. Finally, we connect the multilinear and outer product rank of A to the multilinear and outer product rank of sym(A).