  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.
