Remarks on the Symmetric Rank of Symmetric Tensors

We give sufficient conditions on a symmetric tensor SᵈFⁿ to satisfy the following equality: the symmetric rank of S, denoted as srank\;S, is equal to the rank of S, denoted as rank\;S. This is done by considering the rank of the unfolded S viewed as a matrix A(S). The condition is rank\;S∈\;A(S),rank\;A(S)+1\. In particular, srank\;S=rank\;S for SᵈCⁿ for the cases (d,n)∈\(3,2),(4,2),(3,3)\. We discuss the analogues of the above results for border rank and best approximations of symmetric tensors.