4.19 Tensor Bilinear Form Structure

Tensor Bilinear Form Structure is the body of concepts organizing bilinear forms into a coherent theory: their representation as matrices, their classification by symmetry type and non-degeneracy, the equivalence relation of congruence that governs when two forms are "the same" up to change of basis, and their identification with elements of a tensor product of dual spaces.

The Basic Object

Definition

A bilinear form on vector spaces V and W over a field F is a map

linear in each argument separately. When V = W, f is called a bilinear form on V; when V ≠ W, it is sometimes called a pairing between V and W. Bilinear form structure concerns both cases, with special attention to V = W, where additional symmetry phenomena become possible.

Matrix Representation

Relative to bases {eᵢ} of V and {fⱼ} of W, f is represented by the matrix Aᵢⱼ = f(eᵢ, fⱼ), and every value of f is recovered as f(v,w) = v^T A w. Changing basis on V via a matrix P and on W via a matrix Q transforms the representing matrix by A ↦ P^T A Q, the congruence transformation law governing how the matrix of a bilinear form changes under change of basis.

Classification by Symmetry Type

Symmetric Forms

When V = W, f is symmetric if f(v, w) = f(w, v) for all v, w, equivalently if its matrix satisfies A = A^T. Symmetric bilinear forms correspond to quadratic forms via q(v) = f(v, v), from which f itself can be recovered (in characteristic not 2) by the polarization identity

Alternating and Skew-Symmetric Forms

f is alternating if f(v, v) = 0 for all v, which implies (and, outside characteristic 2, is equivalent to) skew-symmetry, f(v,w) = -f(w,v), equivalently A = -A^T. Non-degenerate alternating forms are called symplectic forms and force V to be even-dimensional.

General Forms

A bilinear form need be neither symmetric nor alternating; any bilinear form decomposes uniquely, in characteristic not 2, into a symmetric part and an alternating part:

matching the decomposition of its matrix into symmetric and skew-symmetric pieces, A = (A + A^T)/2 + (A - A^T)/2.

Non-Degeneracy and Rank

Radical and Non-Degeneracy

The radical of a bilinear form on V is {v ∈ V : f(v, w) = 0 for all w ∈ V}, and f is non-degenerate exactly when its radical is zero, equivalently when its matrix A is invertible. Non-degeneracy is equivalent to the induced linear map V → V*, v ↦ f(v, ·), being an isomorphism.

Rank

The rank of a bilinear form is the rank of its matrix A, equal to dim(V) - dim(radical). Rank is invariant under congruence transformations A ↦ P^T A P, since congruence by invertible matrices preserves rank, making rank a genuine invariant of the form rather than of any particular matrix representative.

Classification Up to Congruence

Congruence as the Governing Equivalence

Two matrices A and A' represent the same bilinear form in different bases exactly when A' = P^T A P for some invertible P; this relation, congruence, replaces matrix similarity as the relevant notion of equivalence for bilinear forms, reflecting that a bilinear form transforms via two copies of the change-of-basis matrix, one on each side, rather than one matrix and its inverse.

Classification of Symmetric Forms

Over an algebraically closed field, every symmetric bilinear form is congruent to a diagonal matrix with entries 0 or 1, so rank is a complete invariant. Over the real numbers, Sylvester's law of inertia refines this: every real symmetric bilinear form is congruent to a diagonal matrix with entries +1, -1, and 0, and the number of each is an invariant, the signature, giving a complete classification up to congruence.

Classification of Alternating Forms

Every alternating bilinear form is congruent to a block-diagonal matrix built from 2 × 2 blocks representing a standard symplectic pairing, plus zero blocks accounting for the radical; consequently rank alone, which is always even for an alternating form, is a complete invariant up to congruence.

Relation to the Tensor Product

Bilinear Forms as Elements of a Dual Tensor Product

By the universal property of the tensor product, bilinear forms on V × W correspond to linear functionals on V ⊗ W, and in the finite-dimensional case to elements of V* ⊗ W*. This identification recasts every classification question about bilinear forms, symmetry, rank, congruence class, as a question about elements of a specific dual tensor product space, connecting bilinear form structure directly to tensor algebra.

Symmetric and Alternating Parts as Tensor Subspaces

The symmetric and alternating bilinear forms on V correspond, under this identification, to the subspaces Sym²(V*) and ⋀²(V*) of V* ⊗ V*, the images of the symmetrization and antisymmetrization projections applied to V* ⊗ V*; the decomposition of a general bilinear form into symmetric and alternating parts is the decomposition of V* ⊗ V* itself into these two complementary subspaces (in characteristic not 2).

Operations on Bilinear Forms

Direct Sum

Given bilinear forms f on V and g on V', their direct sum f ⊕ g on V ⊕ V' is defined by (f ⊕ g)((v,v'),(w,w')) = f(v,w) + g(v',w'), with matrix the block-diagonal combination of the two matrices; this operation preserves symmetry type and adds ranks and, over the reals, adds signatures componentwise.

Restriction to a Subspace

Restricting a bilinear form on V to a subspace U ⊆ V produces a bilinear form on U, whose rank can only decrease or stay the same; a subspace on which the restricted form vanishes identically is called isotropic, a notion central to the classification of quadratic and symplectic forms.

Applications Driven by This Structure

Geometry

Symmetric bilinear forms with definite signature define inner products, giving rise to Euclidean and Riemannian geometry; indefinite signatures give rise to pseudo-Euclidean geometries, including the Minkowski form of special relativity, whose signature (+,-,-,-) or (-,+,+,+) is a direct invariant of the associated bilinear form.

Mechanics

Non-degenerate alternating bilinear forms, symplectic forms, structure the phase space of Hamiltonian mechanics, with the antisymmetry of the form encoding the antisymmetry of the Poisson bracket between observables.