4.2.4 Tensor Bilinear Scalar Output Case

The Tensor Bilinear Scalar Output Case explores how bilinear forms produce scalar outputs from tensor pairs, central to algebraic structures and linear transformations.

Tensor Bilinear Scalar Output Case is the specialization of a bilinear map in which the codomain is the ground field itself rather than an arbitrary vector space, so that the map assigns a single scalar to every ordered pair of vectors while remaining linear in each argument separately. Formally this is a map $B : V \times W \to F$ where $F$ is the scalar field, and such maps are called bilinear forms. This case is the most heavily used instance of bilinearity in tensor algebra because bilinear forms are exactly the objects identified with elements of the tensor product of the two dual spaces, and they include inner products, symplectic forms, and the canonical pairing between a vector space and its dual.

Defining Properties

Bilinearity with Scalar Codomain

A scalar-valued bilinear map satisfies, for fixed second argument, linearity in the first argument:

B (α v_{1} + β v_{2}, w) = α B (v_{1}, w) + β B (v_{2}, w)

and the symmetric condition holds in the second argument. Because the output $α$ lies in the same field as the scalars multiplying the vectors, no extra structure on the codomain is needed to state these conditions, which is why the scalar output case is the simplest and most foundational bilinear setting.

Identification with the Dual Tensor Product

Every scalar-valued bilinear map on $V \times W$ corresponds uniquely to an element of $V^{\otimes W^{}}$ , the tensor product of the dual spaces. If ${e^{i}}$ and ${f^{j}}$ are the dual bases, then

B = \sum_{i, j} b_{i j} e^{i} \otimes f^{j}

with $b_{i j} = B (e_{i}, f_{j})$ , exhibiting the scalar-valued bilinear map as a rank-two covariant tensor.

Matrix Representation

The Gram-Style Matrix

In finite dimensions the scalar output case is entirely captured by a matrix $B = (b_{i j})$ so that

B (v, w) = v^{T} B w

a single real or complex number for every pair of coordinate vectors, in contrast to the vector-valued case where the analogous expression would return a vector rather than a scalar.

Special Subtypes

When $V = W$ , the scalar output case specializes further into symmetric forms, for which $B (v, w) = B (w, v)$ , and antisymmetric or alternating forms, for which $B (v, w) = - B (w, v)$ . Symmetric forms correspond to symmetric matrices and include all inner products; alternating forms correspond to skew-symmetric matrices and include the symplectic forms underlying Hamiltonian mechanics.

Canonical Examples

The Dual Pairing

The evaluation pairing between a vector space and its dual, $⟨ \cdot, \cdot ⟩ : V^{*} \times V \to F$ , given by $⟨ φ, v ⟩ = φ (v)$ , is the archetypal scalar-valued bilinear map, and it is this pairing that identifies the dual of a tensor product with the tensor product of duals.

Inner Products and Metric Tensors

A positive-definite symmetric bilinear form is an inner product, and on a smooth manifold a smoothly varying, nondegenerate, symmetric scalar-valued bilinear form on each tangent space is a Riemannian metric tensor; both are scalar output cases whose coefficients $g_{i j}$ form the metric tensor components used throughout differential geometry.

Distinction from the Vector Output Case

Why the Scalar Case Is Structurally Special

Unlike a general vector-valued bilinear map, a scalar-valued one has no free index remaining in its output, so its entire information content is packed into the two-index coefficient array $b_{i j}$ . This makes the scalar output case the natural home for rank-two covariant tensors, whereas a vector-valued bilinear map corresponds to a tensor of type $(1, 2)$ , carrying an extra contravariant index for the output.

Degeneracy and Rank

A scalar-valued bilinear form is called nondegenerate when the induced linear map $v \mapsto B (v, \cdot)$ from $V$ to $W^{*}$ is an isomorphism, equivalently when the matrix $B$ is invertible; nondegenerate scalar-valued bilinear forms are precisely those used to raise and lower indices in tensor calculus.