# Symmetric Tensors and Symmetric Tensor Rank

```@article{Comon2008SymmetricTA,
title={Symmetric Tensors and Symmetric Tensor Rank},
author={Pierre Comon and Gene H. Golub and Lek-Heng Lim and Bernard Mourrain},
journal={SIAM J. Matrix Anal. Appl.},
year={2008},
volume={30},
pages={1254-1279}
}```
• Published 12 February 2008
• Mathematics, Computer Science
• SIAM J. Matrix Anal. Appl.
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\$ nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of which is symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary…
561 Citations

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It is argued that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rank-r approximations, and a natural way of overcoming the ill-posedness of the low-rank approximation problem is proposed by using weak solutions when true solutions do not exist.
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