I'm not quite sure about the Tensorial interpretation but the closest rank-1 approximation to the matrix is essentially the outer-product of the two dominant singular vectors amplified by the singular value.
In simple words, if [U E V] = svd(X)
, then the closest rank-1 approximation to X
is the outer-product of the first singular vectors multiplied by the first singular value.
In MATLAB, you could do this as:
U(:,1)*E(1,1)*V(:,1)'
Which yields:
ans =
2.0752 1.4487 2.7017
3.9606 2.7649 5.1563
Also, mathematically speaking, the kronecker product of a row vector and a column vector is essentially their outer product. So, you could do the same thing using Kronecker products as:
(kron(U(:,1)',V(:,1))*E(1,1))'
Which yields the same answer.