Remarks on the symmetric rank of symmetric tensors

Abstract

We give sufficient conditions on a symmetric tensor S in S^dF^n to satisfy the 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) is in {rank(A(S)),rank (A(S))+1}. In particular, srank(S)=rank(S) for S in S^dC^n for the cases (d,n) in {(3,2),(4,2),(3,3)}. We discuss the analogs of the above results for border rank and best approximations of symmetric tensors.