14.21.3 Tensor Map Product Form Construction Role

The Tensor Map Product Form Construction Role defines how tensors interact through mappings, shaping algebraic structures in multilinear operations.

Tensor Map Product Form Construction Role is the use of the tensor product of maps to combine a bilinear form on $V$ with a bilinear form on $W$ into a single bilinear form on $V \otimes W$ , by identifying each bilinear form with a linear map into the corresponding dual space and applying the tensor product construction to those linear maps.

Identifying Bilinear Forms With Linear Maps

The Standard Correspondence

A bilinear form $β : V \times V \to K$ corresponds to a linear map $β_{♯} : V \to V^{*}$ defined by

β_{♯} (v) = β (v, -)

sending each vector to the functional obtained by fixing it as the first argument of $β$ , and likewise a bilinear form $γ : W \times W \to K$ corresponds to $γ_{♯} : W \to W^{*}$ .

Applying the Tensor Product Construction

Taking the tensor product of these two linear maps gives

β_{♯} \otimes γ_{♯} : V \otimes W \to V^{*} \otimes W^{*}

which, composed with the canonical identification of $V^{*} \otimes W^{*}$ with ${(V \otimes W)}^{*}$ , produces exactly the map corresponding to a new bilinear form on $V \otimes W$ , denoted $β \otimes γ$ .

The Resulting Form on Simple Tensors

Direct Formula

On simple tensors, the form $β \otimes γ$ is given directly by

(β \otimes γ) (v_{1} \otimes w_{1}, v_{2} \otimes w_{2}) = β (v_{1}, v_{2}) \cdot γ (w_{1}, w_{2})

a product of the two original bilinear forms evaluated on the corresponding factors, matching the intuitive expectation for how a combined form on a tensor product space ought to behave on simple tensors.

Extension by Bilinearity

For general elements of $V \otimes W$ , expressed as sums of simple tensors, the value of $β \otimes γ$ is obtained by the same double-sum expansion used generally for the tensor product of maps, applying bilinearity of the tensor symbol and of $β$ and $γ$ separately in each slot.

Symmetric, Alternating, and Quadratic Cases

Symmetric Forms Combine to a Symmetric Form

If $β$ and $γ$ are both symmetric, meaning $β (a, b) = β (b, a)$ for all $a, b$ , the product formula on simple tensors shows $β \otimes γ$ is symmetric as well, since swapping both pairs of arguments swaps each factor and each factor is individually symmetric.

Associated Quadratic Form

Restricting $β \otimes γ$ to the diagonal gives the associated quadratic form on simple tensors,

Q (v \otimes w) = (β \otimes γ) (v \otimes w, v \otimes w) = β (v, v) \cdot γ (w, w)

giving a direct product formula for the induced quadratic value on a simple tensor in terms of the quadratic values of $β$ and $γ$ on the individual vectors.

Application to Inner Products

Combining Two Inner Products

If $β$ and $γ$ are positive-definite symmetric bilinear forms, that is, inner products, on real vector spaces $V$ and $W$ , the combined form $β \otimes γ$ is symmetric by the argument above, and on an orthonormal basis input element $e_{i} \otimes f_{j}$ , evaluates to $1$ against itself and $0$ against any other basis input element, since the individual bases are orthonormal for $β$ and $γ$ respectively.

The Resulting Structure Is Positive Definite

Because the diagonal entries $β (e_{i}, e_{i}) \cdot γ (f_{j}, f_{j})$ are products of two positive numbers, hence positive, and the basis input elements diagonalize the combined form, $β \otimes γ$ is itself positive definite, exhibiting the tensor map product's form construction role as the standard mechanism for equipping a tensor product of inner product spaces with its own natural inner product.