...$F$¹

One may see the $e_i \otimes f_j$ as the canonical basis elements of the tensor space and represented by the matrices $e_i\;{}^t f_j$ .

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... tensor-²

Note that for tensors of order two (i.e. matrices) it coincides with the rank definition of linear maps.

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... optimisation³

The least square approximation of $A$ up to $r'>r$ orthogonal decomposed tensors, contains the approximation up to $r$, and is the truncation of 11 up to $r'$ terms (with a decreasing order of singular values).

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