14.6.2 Tensor Bilinear Form Product Argument Pairing

Tensor Bilinear Form Product Argument Pairing explains how bilinear forms act on tensor products, creating pairings through algebraic structures in multilinear algebra.

Tensor Bilinear Form Product Argument Pairing is the specific reorganization of the four spaces involved in combining two bilinear forms, regrouping the first argument of each original form into one tensor pair and the second argument of each into another, so the combined map can be read as a bilinear form on this new pairing rather than on the pairing produced immediately by the general tensor product of maps.

The Need for Reassociation

Mismatch Between the Raw Combination and the Desired Pairing

Combining bilinear forms b on V times W and c on U times X through the general tensor product of maps, applied to their linear incarnations, produces a map on

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

which groups the first argument of b with the entire pair from c, rather than pairing the first arguments of b and c together and the second arguments together, so a reassociation is required before the combined map can be regarded as a bilinear form on V tensor U and W tensor X.

The Reassociation Isomorphism

The argument pairing is achieved through the canonical isomorphism

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

built from the associativity and symmetry of the tensor product, which permutes the middle two factors so that V is regrouped with U, and W is regrouped with X, matching the desired argument pairing.

Effect of the Argument Pairing on Elementary Tensors

Regrouping Elementary Tensors

On elementary tensors, the argument pairing sends

(v \otimes w) \otimes (u \otimes x) \mapsto (v \otimes u) \otimes (w \otimes x),

extracting the first component of each original elementary tensor into the new first tensor pair, and the second component of each into the new second tensor pair, exactly the reorganization required to interpret the resulting map as a bilinear form on (V tensor U) times (W tensor X).

The Combined Bilinear Form After Pairing

After applying the argument pairing, the combined form acts on a pair from V tensor U and W tensor X by

d (v \otimes u, w \otimes x) = b (v, w) \cdot c (u, x),

recovering, after the reassociation, exactly the intended product bilinear form built from evaluating b and c separately and multiplying their scalar outputs.

Consequences for the Gram Matrix

Matching the Kronecker Product Ordering

The argument pairing is precisely what justifies describing the combined Gram matrix as the Kronecker product of the Gram matrix of b and the Gram matrix of c with respect to bases of V tensor U and W tensor X built from interleaving basis vectors of V with U and of W with X in the order dictated by the pairing.

Necessity of Consistent Basis Ordering

Because the argument pairing fixes a specific correspondence between the reorganized tensor factors and the original spaces, any computation of the combined Gram matrix must use bases of V tensor U and W tensor X that respect this same correspondence, since using a different ordering convention would produce a permuted matrix rather than the Kronecker product as normally stated.

Argument Pairing Under Iterated Combination

Extending to More Than Two Forms

When combining more than two bilinear forms, the argument pairing generalizes to a permutation regrouping all first arguments together and all second arguments together across every form involved, and this generalized pairing is built by repeated application of the two-form reassociation isomorphism, applied pairwise until every original first argument sits in a single combined first-argument space and every original second argument sits in a single combined second-argument space.