  Canonical Polyadic Decomposition (CPD) of a third-order tensor is
decomposition in a minimal number of rank-$1$ tensors. We call an algorithm
algebraic if it is guaranteed to find the decomposition when it is exact and if
it only relies on standard linear algebra (essentially sets of linear equations
and matrix factorizations). The known algebraic algorithms for the computation
of the CPD are limited to cases where at least one of the factor matrices has
full column rank. In the paper we present an algebraic algorithm for the
computation of the CPD in cases where none of the factor matrices has full
column rank. In particular, we show that if the famous Kruskal condition holds,
then the CPD can be found algebraically.
