14.6 Tensor Bilinear Form Product Structure

Tensor Bilinear Form Product Structure is the organization of the data specific to the bilinear form case of the tensor product of maps, in which the maps involved are linear incarnations of bilinear forms on a tensor product, together with the identifications and combination rules this specialization supports.

Specializing the General Structure to Bilinear Forms

Bilinear Forms as Linear Maps on a Tensor Product

Where the general structure treats maps between arbitrary spaces, the bilinear form product structure restricts attention to linear maps of the type

each arising, through the universal property, from an ordinary bilinear form b on V times W, so the bilinear form product structure treats V tensor W itself as the domain on which its maps act, rather than treating V and W as two separate domains joined only through tensoring the maps.

Two Bilinear Forms Combined

Given a second bilinear form c on U times X with linear incarnation c-tilde, the bilinear form product structure combines b-tilde and c-tilde through the general tensor product of maps, producing a linear map on

which, after reassociating and permuting factors, corresponds to a combined bilinear form on V tensor U and W tensor X.

Internal Assembly of the Structure

From Two Bilinear Forms to a Combined Form

The bilinear form product structure assembles the pair b and c by first passing each to its linear incarnation, tensoring these two linear maps, and then reinterpreting the result as a bilinear form on the reassociated pair of spaces, giving a three-step process: linearize, tensor, then reinterpret as bilinear.

Gram Matrix Assembly

With bases fixed, the bilinear form product structure is realized concretely by forming the Kronecker product of the Gram matrices of b and c, giving the Gram matrix of the combined form directly, and bypassing the abstract linearization and reinterpretation steps once matrices are available.

Preserved Invariants Within the Structure

Rank Multiplicativity

The bilinear form product structure preserves the multiplicative rank property already established for tensor products of maps in general: the rank of the combined form equals the product of the rank of b and the rank of c, since rank is computed identically whether a bilinear form is regarded directly or through its linear incarnation.

Symmetry and Alternation Type

The structure tracks how the symmetry type of the combined form depends on the symmetry types of b and c: two symmetric forms combine to a symmetric form, while combining a symmetric form with an alternating form produces an alternating form, mirroring the sign rules governing products of symmetric and alternating multilinear objects generally.

Structural Compatibility with Composition

Precomposition by Operators

If f is an operator on V and g is an operator on W, the bilinear form product structure is compatible with precomposition, since the linear incarnation of the pulled-back form b(f(v), g(w)) equals b-tilde composed with f tensor g, matching the composition identity of tensor products of maps specialized to this bilinear setting.

Compatibility with the Functional Product Structure

When b and c both have rank one, arising as elementary tensors of individual functionals, the bilinear form product structure reduces exactly to the functional product structure, so the general bilinear form product structure can be seen as the closure of the rank-one functional case under finite sums, extending the functional product structure to bilinear forms of arbitrary rank.