14.6.1 Tensor Bilinear Form Product Factor Selection

Tensor Bilinear Form Product Factor Selection is the process of choosing which bilinear form acts on which pair of tensor factors when combining two bilinear forms through the tensor product of maps, together with the reassociation needed to interpret the combined map as a bilinear form on a new pairing of spaces.

The Basic Selection

Choosing a Bilinear Form per Argument Pair

Given a bilinear form b on V times W and a second bilinear form c on U times X, factor selection assigns b to the pair (V, W) and c to the pair (U, X), producing, through the general tensor product of maps applied to their linear incarnations, a linear map on

Selecting c for the first pair and b for the second instead produces a differently ordered combined map, matching the general sensitivity of tensor products of maps to the order of the factor map pair.

Reassociation After Selection

Because the combined map initially lands in a tensor product of two tensor products rather than in a tensor product of the four original spaces reorganized as V tensor U and W tensor X, factor selection for bilinear forms includes a reassociation step, permuting and regrouping the four tensor factors so the combined map can be reinterpreted directly as a bilinear form on the new pair (V tensor U, W tensor X).

Consequences of the Selection

Selection Determining the Combined Rank

The factor selection made for b and c determines the rank of the combined bilinear form as the product of the individual ranks of b and c, so selecting forms of low rank, such as rank-one forms built from single functionals, produces a combined form whose rank is correspondingly small, regardless of the dimensions of the ambient spaces.

Selection of One Form as an Inner Product

A common factor selection takes b to be a fixed inner product on V, held constant, while c ranges over a family of bilinear forms on varying pairs U times X; this selection produces a family of combined forms on V tensor U and V tensor X differing only through the varying second selection, isolating the effect of changing c while the contribution of b remains fixed.

Selection and Symmetric or Alternating Structure

Selection Preserving Symmetry

If both b and c are selected to be symmetric bilinear forms, the reassociated combined form is symmetric under the simultaneous exchange of V with W and U with X, so a factor selection of two symmetric forms is guaranteed to produce a symmetric combined form without further verification.

Selection Mixing Symmetric and Alternating Forms

If one factor selection is symmetric while the other is alternating, the combined form acquires the alternating sign under exchange contributed by the alternating factor, so the factor selection alone determines the symmetry type of the result, following the standard sign rule for combining a symmetric object with an alternating one.

Selection in the Matrix Description

Kronecker Product of Gram Matrices

Once bases are fixed and factor selection has been made, the Gram matrix of the combined form is the Kronecker product of the Gram matrix of b, corresponding to the selection of b for the first pair, and the Gram matrix of c, corresponding to the selection of c for the second pair, giving an explicit and directly computable outcome of the selection.

Selection Order and Matrix Ordering

Because the Kronecker product is sensitive to the order of its two matrix arguments, the order in which b and c are selected for the first and second pair directly determines the block ordering of the resulting Gram matrix, so reversing the factor selection produces a permuted version of the same combined Gram matrix rather than an entirely different matrix.