14.8.5 Tensor Map Product Codomain Ordering

Tensor Map Product Codomain Ordering is the fixed sequence in which the two codomain factor spaces appear within the codomain tensor product W1 tensor W2, determining which factor receives the image of which map, and thereby which output component appears first and which second in every elementary tensor produced by the construction.

The Role of Ordering in the Codomain

Ordering Distinguishes the Two Factors

The codomain factor spaces W1 and W2, though possibly of the same dimension or even literally the same space, are distinguished within the codomain tensor product by their position: W1 occupies the first slot, receiving the image of f, and W2 occupies the second slot, receiving the image of g, and this positional distinction is what allows the output f(v) tensor g(w) to be unambiguous.

Consequence of Swapping the Order

Swapping the codomain ordering, so that W2 is treated as first and W1 as second, produces the tensor product W2 tensor W1, canonically isomorphic to W1 tensor W2 through the swap map sending w1 tensor w2 to w2 tensor w1, but not identical to it as a constructed object; a tensor product of maps producing outputs in one ordering does not automatically produce outputs in the other ordering without composing with this swap isomorphism.

Ordering and the Elementary Output Rule

Ordering Fixes Which Image Appears First

The codomain ordering is what makes the elementary output rule

unambiguous: f(v) is understood to occupy the first slot of the output, matching the codomain ordering that places W1 first, and g(w) the second slot, matching W2 second.

Effect of Reordering on Recognizing Equal Outputs

If a different construction produces an output g(w) tensor f(v) instead, lying in W2 tensor W1 rather than W1 tensor W2, comparing this output to f(v) tensor g(w) requires first applying the swap isomorphism to bring both outputs into a common codomain ordering, since the two elementary tensors, though built from the same two vectors, are elements of formally different tensor product spaces prior to any such identification.

Ordering in the Kronecker Product Representation

Ordering Fixes the Row Block Structure

Once bases are chosen respecting the codomain ordering, the rows of the Kronecker product matrix A tensor B are grouped into blocks corresponding to the codomain ordering, with each block of rows corresponding to a single basis vector of W1 tensored with the entire basis of W2; reversing the codomain ordering would correspond instead to the matrix B tensor A, related to A tensor B by a row permutation given by a commutation matrix on the output side.

Consistency Between Domain and Codomain Ordering

For the Kronecker product formula to correctly represent f tensor g, the codomain ordering used to index the rows of the matrix must be chosen consistently with the domain ordering used to index its columns, since a mismatch between these two orderings would produce a matrix representing a differently permuted map rather than f tensor g itself.

Ordering Under Composition and Extension

Ordering Preserved Under Composition

When composing tensor products of maps, the codomain ordering of the composite matches the codomain ordering of the second tensor product of maps in the chain, since composition transforms values already assigned to each ordered factor without permuting the factors of the codomain tensor space itself.

Ordering and Single-Factor Extensions

The codomain ordering also determines which single-factor extension is meant by an expression such as f tensor identity: because the codomain ordering fixes W1 as first and W2 as second, this expression unambiguously places the image of f in the first slot of the output while the identity leaves the second slot occupied by the unchanged second component, and reversing the codomain ordering would require writing the extension the other way around to preserve the same intended meaning.