14.6.4 Tensor Bilinear Form Product Component Rule

The Tensor Bilinear Form Product Component Rule explains how tensor components combine under bilinear operations in multilinear algebra.

Tensor Bilinear Form Product Component Rule is the explicit formula giving the scalar value of a combined bilinear form on a pair of elementary tensors, expressing that value as the ordinary product of the two original bilinear forms evaluated separately on their respective components.

Statement of the Component Rule

The Defining Formula

For bilinear forms b on V times W and c on U times X, the component rule states that the combined form d on (V tensor U) times (W tensor X), obtained after the argument pairing, satisfies

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

on elementary tensors v tensor u drawn from V tensor U and w tensor x drawn from W tensor X, giving the value of the combined form as a single ordinary scalar product of the two component evaluations.

Well-Definedness of the Component Rule

Because the right-hand side is bilinear in each of v, w, u, and x separately, the component rule extends consistently to a well-defined bilinear form on the full tensor product spaces, via the same universal-property argument used to establish well-definedness for tensor products of maps in general, so the component rule need not be checked separately for consistency once bilinearity of both b and c is known.

Extension to General Elements

Sums of Elementary Tensors

For general elements expressed as finite sums of elementary tensors, the component rule extends by bilinearity to

d (\sum_{i} v_{i} \otimes u_{i}, \sum_{j} w_{j} \otimes x_{j}) = \sum_{i, j} b (v_{i}, w_{j}) \cdot c (u_{i}, x_{j}),

a double sum over every pair of terms drawn from the two decompositions, reducing the evaluation of the combined form on general elements to a finite collection of ordinary evaluations of b and c.

Coordinate Form of the Component Rule

Gram Matrix Entries

If B and C denote the Gram matrices of b and c with respect to fixed bases, the component rule states, entry by entry, that the (i,k),(j,l) entry of the combined Gram matrix, indexed by pairs of basis vectors from V tensor U and W tensor X, equals

b_{ij} \cdot c_{kl},

matching exactly the entry pattern of the Kronecker product B tensor C, so the component rule is the entrywise statement underlying the Kronecker product description of the combined Gram matrix.

Consistency with the Kronecker Product

The component rule, applied to every pair of basis vectors, reconstructs the full Kronecker product matrix B tensor C from its individual entries, confirming that the abstract component rule and the concrete Kronecker product description of the combined Gram matrix carry exactly the same information, presented at two different levels of explicitness.

Special Cases of the Component Rule

Component Rule When One Form Is an Inner Product

If c is chosen to be the standard inner product on U equal to X, the component rule reduces to

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

showing that the component rule reduces exactly to scaling b by the ordinary inner product of the second components whenever c is chosen this way.

Component Rule for Repeated Combination

Applying the component rule twice, once to combine b with c and again to combine the result with a third form e, gives the same final value as combining c with e first and then with b, since the ordinary scalar multiplication appearing on the right-hand side of the component rule is associative and commutative, so the order of iterated combination via the component rule does not affect the final scalar values produced.