Remarks On The Symmetric Rank Of Symmetric Tensors

We give sufficient conditions on a symmetric tensor S ∈ 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∈{rank A(S), rank A(S)+1}. In particular, srank S = rank S for S ∈ 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.