4.19.1 Tensor Bilinear Form Scalar Output

Tensor Bilinear Form Scalar Output is the property of a bilinear form f: V × W → F that its result, for any pair of input vectors, is a single element of the base field F rather than a vector in some other space. Scalar output is what distinguishes a bilinear form proper from a general bilinear map V × W → U into an arbitrary vector space U, and it is this restriction that allows bilinear forms to be organized into matrices, paired with dual spaces, and used to define notions such as orthogonality and non-degeneracy.

The Defining Condition

Two Slots, One Number

A bilinear form takes exactly two vector arguments and combines them into a scalar, linearly in each argument separately:

and symmetrically for the second argument. Because the result lands in F, sums and scalar multiples of the outputs are computed within the field itself, without needing any additional vector space structure to interpret them.

Distinguishing Forms From General Bilinear Maps

A bilinear map g: V × W → U with U an arbitrary vector space is not automatically a bilinear form unless U = F. The term "form" is reserved specifically for the scalar-valued case; a general bilinear map into a higher-dimensional U, such as matrix multiplication Hom(V,U) × Hom(U,W) → Hom(V,W), is bilinear but not a bilinear form.

Matrix Representation

The Matrix of Scalars

If V and W are finite-dimensional with bases {eᵢ} and {fⱼ}, the scalar output at every pair of basis vectors defines a matrix A with entries Aᵢⱼ = f(eᵢ, fⱼ), and the scalar value of f on any pair of vectors is recovered by

This single-number output at each entry is exactly what allows the entire bilinear form to be encoded in an ordinary matrix, in contrast to a vector-valued bilinear map, which would require a matrix whose entries are themselves vectors.

Symmetric, Skew, and Non-Degenerate Cases

When V = W, the matrix A may be symmetric (A = A^T), giving a symmetric bilinear form with f(v,w) = f(w,v); or skew-symmetric (A = -A^T), giving an alternating bilinear form with f(v,w) = -f(w,v). A bilinear form is called non-degenerate if its matrix A is invertible, equivalently if no nonzero v satisfies f(v, w) = 0 for all w.

The Induced Linear Map to the Dual

From Bilinear Form to Linear Map

Scalar output allows a bilinear form f: V × W → F to be reinterpreted, by fixing one argument, as a linear map into a dual space:

This construction relies essentially on f(v, w) being a scalar, since a linear functional on W must itself return a scalar; if f returned vectors in a general U instead, fixing v would produce a linear map W → U, not an element of W*. Non-degeneracy of f corresponds exactly to f̃ being injective, and when V and W have equal finite dimension, to f̃ being an isomorphism.

Canonical Examples

Inner Products

A real inner product ⟨·,·⟩: V × V → R is a symmetric bilinear form with scalar output satisfying positive-definiteness, and its associated matrix in an orthonormal basis is the identity; the scalar nature of the output is what allows lengths √⟨v,v⟩ and angles cos θ = ⟨v,w⟩ / (‖v‖‖w‖) to be defined directly from its values.

Symplectic Forms

An alternating, non-degenerate bilinear form ω: V × V → F is called symplectic; its scalar output at a pair of vectors measures a signed area-like quantity, and symplectic forms underlie the phase-space structure of Hamiltonian mechanics, where the scalar value ω(v, w) for tangent vectors v, w determines the Poisson bracket of associated functions.

The Trace Form on Matrices

On the space of n × n matrices, f(A, B) = tr(AB) is a symmetric bilinear form with scalar output, non-degenerate, and used to identify the space of matrices with its own dual space via the induced map A ↦ tr(A · -).

Relation to the Tensor Product

Scalar-Valued Bilinear Forms as Functionals on V ⊗ W

By the universal property of the tensor product specialized to two factors, every bilinear form f: V × W → F corresponds to a unique linear functional f̃ ∈ (V ⊗ W)*. In the finite-dimensional case, (V ⊗ W)* ≅ V* ⊗ W*, so bilinear forms with scalar output correspond exactly to elements of V* ⊗ W*, with an elementary tensor φ ⊗ ψ corresponding to the rank-one form (v, w) ↦ φ(v)ψ(w).

Rank as Minimal Decomposition

A scalar-valued bilinear form's matrix rank equals the minimal number of rank-one product forms φᵢ(v)ψᵢ(w) needed to write it as a sum, matching the minimal number of elementary tensors of dual vectors needed to express the corresponding element of V* ⊗ W*; this identification of matrix rank with tensor rank is a direct consequence of the scalar-output restriction, since it is precisely what makes the correspondence with V* ⊗ W* available.

Why Scalar Output Enables Additional Structure

Multiplicativity and Powers

Scalar-valued bilinear forms can be raised to powers, composed with other scalar-valued forms, or combined multiplicatively in ways vector-valued bilinear maps cannot: for instance, a quadratic form q(v) = f(v,v) derived from a symmetric bilinear form with scalar output can itself be squared, added to other quadratic forms, or examined for definiteness, all operations relying on the output being an ordinary number in F.

Determinant and Discriminant

Because the matrix A representing a scalar-valued bilinear form is an ordinary matrix over F, its determinant, the discriminant of the form, is defined and used to test non-degeneracy and, over the reals, to classify the form's signature; no analogous scalar invariant is directly available for a bilinear map whose output is vector-valued rather than scalar-valued.