In the tensor space SymdR2 of binary forms we study the best rank k approximation
problem. The critical points of the best rank 1 approximation problem are the eigenvectors
and it is known that they span a hyperplane. We prove that the critical points of the best rank
k approximation problem lie in the same hyperplane. As a consequence, every binary form
may be written as linear combination of its critical rank 1 tensors, which extends the Spectral
Theorem from quadratic forms to binary forms of any degree. In the same vein, also the best
rank k approximation may be written as a linear combination of the critical rank 1 tensors,
which extends the Eckart–Young theorem from matrices to binary forms.
