14.2.3 Tensor Bilinear Form Product Area

The Tensor Bilinear Form Product Area examines how bilinear forms interact with tensor products, revealing algebraic structures in multilinear algebra.

Tensor Bilinear Form Product Area is the branch of study concerned with tensor products of linear maps in the case where the maps involved are themselves bilinear forms, treated as linear maps out of a tensor product, and with how such forms combine when their two argument spaces are tensored together.

Bilinear Forms as Linear Maps on a Tensor Product

Reformulating a Bilinear Form

A bilinear form b on the pair of spaces V and W, ordinarily presented as a map

b : V \times W \to F

corresponds, through the universal property of the tensor product, to a unique linear map

\tilde{b} : V \otimes W \to F

satisfying b-tilde of v tensor w equal to b of v and w. The bilinear form product area studies bilinear forms specifically through this linear-map incarnation, so that constructions applicable to tensor products of maps in general become directly applicable to bilinear forms.

Tensoring Two Bilinear Forms

Given a second bilinear form c on a pair of spaces U and X, the tensor product of the corresponding linear maps b-tilde and c-tilde produces a linear map on

(V \otimes W) \otimes (U \otimes X),

which, after reassociating and permuting the tensor factors, corresponds to a bilinear form on the pair formed by V tensor U and W tensor X, giving a systematic way to combine two bilinear forms into a single bilinear form on tensored argument spaces.

Compatibility with Classical Invariants

Rank of a Bilinear Form

The rank of a bilinear form, defined as the rank of its associated linear map into the dual of one of the argument spaces, behaves multiplicatively under this tensoring operation: the rank of the combined form on V tensor U and W tensor X equals the product of the rank of b and the rank of c, mirroring the rank multiplicativity already established for tensor products of maps in general.

Symmetric and Alternating Forms

If b and c are both symmetric bilinear forms, the combined form obtained from their tensor product is again symmetric under the simultaneous exchange of the two tensored argument pairs; if one of b or c is alternating while the other is symmetric, the parity of the combined form follows the same sign rule as the product of a symmetric and an alternating multilinear object, so that exactly one sign change occurs under the relevant transposition.

Matrix Description

Gram Matrix as Kronecker Product

When V, W, U, X are finite-dimensional with fixed bases, and the Gram matrices of b and c with respect to those bases are B and C respectively, the Gram matrix of the combined bilinear form on V tensor U and W tensor X, with respect to the induced bases of elementary tensors, is the Kronecker product

B \otimes C,

giving an explicit and directly computable matrix representation of the combined form once the individual Gram matrices are known.

Positive Definiteness

If b and c are both positive definite bilinear forms on real vector spaces, meaning V equals W and U equals X with b(v,v) and c(u,u) strictly positive for nonzero v and u, then the combined form on V tensor U is also positive definite, since the eigenvalues of its Gram matrix B tensor C are exactly the pairwise products of the positive eigenvalues of B and C, which remain positive.

Connection to Inner Product Spaces

Tensor Product of Inner Products

When b and c are inner products rather than general bilinear forms, the construction above produces the standard tensor product inner product on V tensor U, under which elementary tensors satisfy

⟨ v_{1} \otimes u_{1}, v_{2} \otimes u_{2} ⟩ = ⟨ v_{1}, v_{2} ⟩ \cdot ⟨ u_{1}, u_{2} ⟩ .

This identity is the source of the multiplicative behavior of norms and angles observed when working with tensor products of vectors drawn from inner product spaces, and it underlies the orthogonality of elementary tensors built from orthogonal bases of the two factors.