14.6.3 Tensor Bilinear Form Product Scalar Output

Tensor Bilinear Form Product Scalar Output is the single element of the field of scalars produced by evaluating a combined bilinear form, built from tensoring two bilinear forms, on a specific pair of arguments drawn from the two reassociated tensor product spaces.

Nature of the Scalar Output

A Single Field Element, Not a Tensor

Evaluating the combined form d, built from b and c through the tensor product of maps and the argument pairing, on a pair of elements t from V tensor U and s from W tensor X produces

a single element of the field F, in contrast to the value f(v) tensor g(w) produced by a general tensor product of maps, which lies in a tensor product of codomains rather than directly in the field.

Reduction from an Elementary Tensor of Scalars

The scalar output arises because each bilinear form individually already produces a scalar, so tensoring two bilinear forms and evaluating avoids ever forming an elementary tensor of two separate scalar outputs; instead, the two scalar evaluations b(v,w) and c(u,x) are multiplied directly within F, using the canonical identification of F tensor F with F.

Computing the Scalar Output

On Elementary Tensor Arguments

For elementary tensor arguments v tensor u and w tensor x, the scalar output is given directly by the component rule,

an ordinary product of two scalar values, each itself the scalar output of one of the two original bilinear forms.

On General Arguments

For general arguments expressed as sums of elementary tensors, the scalar output is obtained by summing the scalar outputs of the component rule over every combination of terms appearing in the two decompositions, producing a single finite sum of scalar products that collapses to one field element regardless of how many terms the decompositions contain.

Properties of the Scalar Output

Bilinearity in the Combined Arguments

The scalar output, viewed as a function of t and s, is itself bilinear on (V tensor U) times (W tensor X), inheriting this property directly from the bilinearity of b and c together with the bilinearity guaranteed by the general tensor product of maps construction, so the combined form d is itself a legitimate bilinear form producing scalar outputs in exactly the same sense as b and c individually.

Vanishing Scalar Output

The scalar output vanishes whenever either b(v,w) or c(u,x) vanishes on the elementary tensor components of the arguments, giving an explicit sufficient condition for d to output zero without needing to compute the full sum over a general decomposition.

Scalar Output and Matrix Computation

Quadratic Form via the Gram Matrix

With bases fixed, the scalar output for arguments represented by coordinate vectors p and q is computed as

with B and C the Gram matrices of b and c, giving a single explicit matrix computation that produces the scalar output directly, bypassing the need to work through the component rule term by term.

Consistency Check via Direct Substitution

Substituting elementary tensor coordinates into this matrix formula reproduces exactly the component rule value, confirming that the scalar output computed via the full Kronecker product Gram matrix agrees with the scalar output computed by evaluating b and c separately and multiplying, regardless of which computational route is followed.