14.7.5 Tensor Map Product Domain Ordering

Tensor Map Product Domain Ordering is the fixed sequence in which the two domain factor spaces appear within the domain tensor product V1 tensor V2, determining which factor is regarded as first and which as second, and thereby which map of the factor map pair acts on which position.

The Role of Ordering in the Domain

Ordering Distinguishes the Two Factors

The domain factor spaces V1 and V2, though possibly of the same dimension or even literally the same space, are distinguished within the domain tensor product by their position: V1 occupies the first slot and V2 occupies the second slot, and this positional distinction is what allows the factor map pair to unambiguously assign f to the first factor and g to the second.

Consequence of Swapping the Order

Swapping the domain ordering, so that V2 is treated as first and V1 as second, produces the tensor product V2 tensor V1 rather than V1 tensor V2; although these two spaces are canonically isomorphic through the swap map sending v tensor w to w tensor v, they are not identical as constructed objects, and a tensor product of maps built for one ordering does not automatically apply to the other without composing with this swap isomorphism.

Ordering and the Elementary Output Rule

Ordering Fixes the Elementary Tensor Formula

The domain ordering is what makes the elementary output rule

unambiguous: v is understood to belong to the first domain factor space, matching the domain of f, and w to the second, matching the domain of g, precisely because the domain ordering has already fixed which factor is which.

Effect of Reordering on the Output

Reordering the domain to treat w as first and v as second, giving the elementary tensor w tensor v in V2 tensor V1, would require applying g to the first slot and f to the second, namely g(w) tensor f(v), a different elementary tensor from f(v) tensor g(w) unless F and W1 happen to coincide in a way that makes the two agree after applying the swap isomorphism.

Ordering in the Kronecker Product Representation

Ordering Fixes the Block Structure

Once bases are chosen respecting the domain ordering, the Kronecker product matrix representation A tensor B places the entries of A as block scalars multiplying the whole matrix B, rather than the reverse; reversing the domain ordering would correspond instead to the Kronecker product B tensor A, a matrix related to A tensor B by a fixed permutation of rows and columns but not generally equal to it entrywise.

Ordering and the Commutation Matrix

The relationship between A tensor B and B tensor A, arising from reversing the domain ordering, is mediated by a permutation matrix known as the commutation matrix, which reindexes the basis elementary tensors according to the swap of the domain ordering, converting the Kronecker product taken in one ordering into the Kronecker product taken in the reversed ordering.

Ordering Under Composition and Extension

Ordering Preserved Under Composition

When composing tensor products of maps, the domain ordering of the composite matches the domain ordering of the first tensor product of maps in the chain, since composition does not permute the factors of the domain tensor space, only transforms the values already assigned to each ordered factor.

Ordering and Single-Factor Extensions

The domain ordering also determines which single-factor extension is meant by an expression such as f tensor identity: because the domain ordering fixes V1 as first and V2 as second, this expression unambiguously means f acting on the first factor while the identity leaves the second factor, namely V2, unchanged, and reordering the domain would require writing the extension the other way around to preserve the same intended meaning.