Symmetric tensors and symmetric tensor rank.pdf

SYMMETRIC TENSORS AND SYMMETRIC TENSOR RANK PIERRE COMON?, GENE GOLUB? , LEK-HENG LIM? , AND BERNARD MOURRAIN? Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank-1 order-k tensor is the outer product of k non-zero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1. Key words. Tensors, multiway arrays, outer product decomposition, symmetric outer product decomposition, candecomp, parafac, tensor rank, symmetric rank, symmetric tensor rank, generic symmetric rank, maximal symmetric rank, quantics AMS subject classifications. 15A03, 15A21, 15A72, 15A69, 15A18 1. Introduction. We will be interested in the decomposition of a symmetric tensor into a minimal linear combination of symmetric outer products of vectors (i.e. of the form v ? v ? · · · ? v). We will see that a decomposition of the form A = ∑r i=1 λivi ? vi ? · · · ? vi (1.1) always exists for any symmetric tensor A (over any field). One may regard this as a generalization of the eigenvalue decomposition for symmetric matrices to higher order symmetric tensors. In particular, this will allow us to define a notion of symmetric tensor rank (as th