14.8.3 Tensor Map Product Codomain Element Form

Tensor Map Product Codomain Element Form is the distinction between an output of a tensor product of maps that takes the form of a single elementary tensor and one that takes the form of a general finite sum of elementary tensors, together with the conditions under which each form actually arises as an output.

The Two Basic Forms

Elementary Tensor Output Form

An output of f tensor g is in elementary tensor form when it can be written as a single term

which occurs automatically whenever the input v tensor w is itself in elementary tensor form, since the elementary output rule sends elementary inputs to elementary outputs directly.

General Sum Output Form

An output not expressible as a single elementary tensor is in general sum form,

typically arising when the input to f tensor g is itself in general sum form, so that linear extension of the elementary output rule produces a sum of several elementary tensor outputs rather than a single one.

When Each Form Arises

Elementary Input Guarantees Elementary Output

If the input is a single elementary tensor v tensor w, the output f(v) tensor g(w) is guaranteed to be in elementary tensor form, since this is exactly the value produced by the elementary output rule, requiring no summation of any kind.

General Input Need Not Force General Output

A general sum input can still produce an output in elementary tensor form, even though the input itself is not a single elementary tensor, if the resulting sum of elementary tensor outputs happens to simplify or collapse into a single elementary tensor, a phenomenon that can occur when f or g has a sufficiently small image or when the input terms interact favorably under the elementary output rule.

Codomain Element Form and Rank

Rank Cannot Increase Beyond What Is Produced

The rank of an output, as a tensor in the codomain structure, is at most the number of terms present in the general sum form of the corresponding input, since the elementary output rule maps each input term to at most one output term, so the codomain element form of the output cannot require more terms in its own minimal general sum form than the input required, though it may require fewer.

Rank Reduction Under the Elementary Output Rule

Rank reduction of this kind occurs when f or g fails to be injective on the relevant vectors, causing distinct terms in the general sum form of the input to produce equal or linearly dependent elementary tensors in the codomain structure, allowing the resulting sum to be simplified into fewer terms than were present in the original input.

Codomain Element Form in Coordinates

Coordinate Recognition of Elementary Form

Given the coordinate vector of an output with respect to the induced basis of the codomain structure, the output is in elementary tensor form precisely when the coefficient matrix obtained by reshaping this coordinate vector according to the dimensions of W1 and W2 has rank at most one, matching the general criterion for recognizing elementary tensor form from coordinates.

Practical Determination via the Kronecker Product Matrix

Since the coordinate vector of an output is obtained by applying the Kronecker product matrix of f and g to the coordinate vector of the input, the codomain element form of any given output can be determined directly by computing this matrix product and then checking the rank of the resulting reshaped coefficient matrix, without needing to trace through the original elementary output rule term by term.