14.3.1 Tensor Map Product Factor Map Pair

The Tensor Map Product Factor Map Pair explores how tensor maps interact through factorization, revealing structural relationships in multilinear algebra.

Tensor Map Product Factor Map Pair is the ordered pair of linear maps supplied as the raw data from which a tensor product of maps is built, consisting of one map assigned to the first tensor factor and one map assigned to the second tensor factor.

Identifying the Pair

The Two Maps and Their Order

Given a tensor product of maps written as f tensor g, the factor map pair is the ordered pair

(f, g)

with

f : V_{1} \to W_{1}

g : V_{2} \to W_{2}

The pair is ordered because f is understood to act on the first tensor factor and g on the second, and reversing the order of the pair without also reversing the order of the tensor factors gives a genuinely different map.

Independence of the Two Maps

The two maps forming the factor map pair are chosen independently of one another: neither the domain nor the codomain of f constrains the domain or codomain of g in any way, and no relationship between f and g, such as one being derived from the other, is assumed by the construction. This independence is what allows the tensor product of maps to be defined for any pair of linear maps whatsoever, regardless of how the two maps arose.

Role of the Pair in the Construction

Generating the Bilinear Map

The factor map pair is the input to the bilinear map

β (v, w) = f (v) \otimes g (w),

whose factorization through the tensor product, guaranteed by the universal property, produces the map f tensor g. Every property of f tensor g traces back to a property of this pair, so specifying the factor map pair completely determines the resulting tensor product map.

Recovering the Pair from the Product

In general, a linear map on a tensor product of the form V1 tensor V2 need not arise as f tensor g for any factor map pair, since the space of all linear maps on V1 tensor V2 to W1 tensor W2 is typically much larger than the set of maps expressible as an elementary tensor of a single pair. When a map does arise this way, the factor map pair producing it is unique up to simultaneously rescaling f by a nonzero scalar and g by its reciprocal, since

(λ f) \otimes (\frac{1}{λ} g) = f \otimes g .

Operations on the Pair

Composing Each Component of the Pair

Given a second factor map pair consisting of f prime and g prime with matching domains, the composition rule for tensor products of maps is expressed entirely in terms of composing the pair componentwise,

(f^{'} \otimes g^{'}) \circ (f \otimes g) = (f^{'} \circ f) \otimes (g^{'} \circ g),

showing that composing two factor map pairs componentwise, then tensoring the result, gives the same map as tensoring each pair first and then composing the two resulting tensor products of maps.

Replacing One Component with an Identity

A distinguished special case of the factor map pair occurs when one component is an identity map, giving the pair (f, identity) or (identity, g), which restricts the action of the resulting tensor product map to a single factor while leaving the other factor untouched, as recorded by the single-factor extension formulas of the tensor product map construction.

Consistency Requirements on the Pair

Domain and Codomain Matching with the Ambient Tensor Products

For the factor map pair to determine a map on a specific tensor product V1 tensor V2, the domain of f must coincide exactly with the first factor V1 and the domain of g must coincide exactly with the second factor V2; likewise, for the resulting map to be regarded as landing in a specific tensor product W1 tensor W2, the codomain of f must coincide with W1 and the codomain of g with W2. Any mismatch between the stated domains or codomains of the pair and the tensor factors under consideration prevents the pair from defining a tensor product map on those particular spaces.