14.3 Tensor Map Product Structure

Tensor Map Product Structure is the organization of the data and relations that together constitute a tensor product of linear maps, encompassing the pair of maps supplied as input, the tensor factors those maps are tied to, and the way this data assembles into a single well-defined linear map on a tensor product.

Constituents of the Structure

The Factor Map Pair

At the core of the structure lies the ordered pair of linear maps

supplied independently of one another, with the order of the pair fixing which map is associated with the first tensor factor and which with the second.

The Factor Space Relation

Bound to the factor map pair is the correspondence tying V1 and V2 to the two factors of the domain of the resulting map, and tying W1 and W2 to the two factors of its codomain, so that the domain and codomain of f tensor g are fully determined as

Assembly of the Structure

From Pair and Relation to a Bilinear Map

The structure assembles the factor map pair and the factor space relation into the bilinear map

with v drawn from V1 and w drawn from V2, taking advantage of the factor space relation to know precisely which spaces beta is defined on.

From the Bilinear Map to the Induced Linear Map

The structure is completed by invoking the universal property of the tensor product on beta, which produces the unique linear map f tensor g on V1 tensor V2 agreeing with beta on elementary tensors. This final step is what converts the raw data of the factor map pair and factor space relation into an actual, single, well-defined map rather than merely a rule stated on generators.

Internal Consistency of the Structure

Compatibility Under Composition

The structure remains internally consistent under composition: composing two tensor products of maps whose factor space relations align, factor by factor, at the shared intermediate spaces produces a new tensor product of maps whose factor map pair is the componentwise composite of the original two pairs,

Compatibility Under Identity Extension

The structure is also preserved under extension by identity maps: replacing one component of the factor map pair with an identity map on a factor leaves the factor space relation for that factor unchanged while trivializing the action of the tensor product map on that factor, giving the single-factor extensions

Representation of the Structure in Coordinates

Matrix Description

Once bases are fixed on all spaces involved, the structure is represented concretely by the Kronecker product of the matrices of f and g, converting the abstract factor map pair and factor space relation into a single explicit block matrix that can be computed and manipulated directly.

Recovering the Structure from a Matrix

Conversely, given a matrix that is known to arise as a Kronecker product, the individual factor matrices, and hence the factor map pair together with the factor space relation, can be recovered by identifying the block pattern of the matrix, up to the ambiguity of rescaling one factor and inversely rescaling the other, which is the only indeterminacy inherent in the structure.