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 T 0 ] ) . 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 ) .