Mathematics – Numerical Analysis
Scientific paper
2008-02-12
Mathematics
Numerical Analysis
To appear in the SIAM Journal on Matrix Analysis and Applications
Scientific paper
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 sum of symmetric 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.
Comon Pierre
Golub Gene
Lim Lek-Heng
Mourrain Bernard
No associations
LandOfFree
Symmetric tensors and symmetric tensor rank does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Symmetric tensors and symmetric tensor rank, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Symmetric tensors and symmetric tensor rank will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-108752