  Given a tensor $f$ in a Euclidean tensor space, we are interested in the
critical points of the distance function from $f$ to the set of tensors of rank
at most $k$, which we call the critical rank-at-most-$k$ tensors for $f$. When
$f$ is a matrix, the critical rank-one matrices for $f$ correspond to the
singular pairs of $f$. The critical rank-one tensors for $f$ lie in a linear
subspace $H_f$, the critical space of $f$. Our main result is that, for any
$k$, the critical rank-at-most-$k$ tensors for a sufficiently general $f$ also
lie in the critical space $H_f$. This is the part of Eckart-Young Theorem that
generalizes from matrices to tensors. Moreover, we show that when the tensor
format satisfies the triangle inequalities, the critical space $H_f$ is spanned
by the complex critical rank-one tensors. Since $f$ itself belongs to $H_f$, we
deduce that also $f$ itself is a linear combination of its critical rank-one
tensors.
