4.19.2 Tensor Bilinear Form Matrix Representation

Understanding how bilinear forms are represented using matrices in the context of tensor algebra.

Tensor Bilinear Form Matrix Representation is the assignment of a matrix A to a bilinear form f: V × W → F, relative to chosen bases of V and W, such that the value f(v, w) for any vectors is recovered by the matrix computation v^T A w. The matrix representation converts the abstract, basis-free notion of a bilinear form into a finite grid of numbers that standard linear-algebraic machinery can act on directly.

Constructing the Matrix

Entry-by-Entry Definition

Given bases {e₁, ..., e_m} of V and {f₁, ..., f_n} of W, the matrix representing f is the m × n matrix A with entries

A_{ij} = f (e_{i}, f_{j})

recording the value of f on every pair of basis vectors, one from V and one from W.

Recovering the Form From the Matrix

Writing an arbitrary v ∈ V and w ∈ W in coordinates as v = ∑ᵢ vⁱeᵢ and w = ∑ⱼ wʲfⱼ, bilinearity gives

f (v, w) = \sum i, j v^{i} w^{j} A_{ij} = v^{T} A w

where v and w are treated as column vectors of coordinates. This equation is exact, holding for every pair of vectors, and it fully determines f given A, since it expresses f in terms of nothing but matrix multiplication.

Change of Basis

The Congruence Rule

If the basis of V changes via v = P v' (new coordinates v' related to old coordinates v by matrix P) and the basis of W changes via w = Q w', then

f (v, w) = v^{T} A w = v^{′T} P^{T} A Q w'

so the matrix representing f in the new bases is A' = P^T A Q. This transformation law, congruence when V = W and Q = P, is distinct from the similarity transformation A' = P^{-1}AP used for linear operators, reflecting that a bilinear form's matrix transforms with two applications of the change-of-basis data rather than one application and its inverse.

Why Two Copies of the Change-of-Basis Matrix Appear

The matrix representation of f involves the coordinates of two separate vectors, v and w, each subject to its own coordinate change; both changes must be undone to express f correctly in the new coordinates, which is why P^T appears on one side and Q on the other, rather than P^{-1} and Q as would occur for a linear map represented by a single matrix acting on one vector.

The Case V = W: Symmetric and Skew Matrices

Symmetry Reflected in the Matrix

When V = W, f is symmetric exactly when A = A^T, and alternating (equivalently, skew-symmetric outside characteristic 2) exactly when A = -A^T; these matrix conditions are basis-independent in the sense that congruence A ↦ P^T A P preserves both A = A^T and A = -A^T, so the symmetry type of f can be checked in any single basis and holds in every basis.

Rank of the Matrix as an Invariant of the Form

The rank of the matrix A equals dim(V) - dim(radical of f) and is preserved under congruence, since P^T A P has the same rank as A whenever P is invertible; matrix rank is therefore an intrinsic invariant of the bilinear form, independent of which basis was used to compute the matrix.

Special Matrix Representations

Diagonal Representation of Symmetric Forms

Every symmetric bilinear form on a finite-dimensional space over a field of characteristic not 2 has a basis in which its matrix is diagonal, obtained by successively choosing basis vectors orthogonal with respect to f to the previously chosen ones, an adaptation of the Gram–Schmidt procedure to a general (possibly indefinite) symmetric form rather than a fixed positive-definite inner product.

Standard Block Representation of Alternating Forms

Every alternating bilinear form has a basis in which its matrix is a direct sum of 2×2 blocks of the standard symplectic form together with a zero block for the radical, giving the simplest possible matrix representation available once the form's rank is known.

Identity Matrix for a Chosen Inner Product

An inner product on a finite-dimensional real vector space has, by definition of an orthonormal basis, the identity matrix as its representation in that basis; other bases give other, generally non-identity, symmetric positive-definite matrices representing the same inner product.

Computational Uses of the Matrix

Evaluating the Form Efficiently

Once A is known, evaluating f(v, w) for any pair of vectors reduces to two matrix-vector multiplications and a dot product, v^T(Aw), an efficient computation directly usable in numerical linear algebra software, independent of any abstract description of f.

Determining Non-Degeneracy and Signature

Non-degeneracy of f is read off from invertibility of A, checked via its determinant; over the real numbers, the signature of a symmetric f is computed from the signs of the eigenvalues of any symmetric matrix representative, or equivalently from the signs appearing when A is reduced to diagonal form via congruence, giving a fully matrix-based route to classifying the form.

Distinguishing From the Operator Matrix

The matrix representation of a bilinear form should not be conflated with the matrix representing a linear operator on the same space: the two obey different transformation laws under change of basis, congruence for the bilinear form versus similarity for the operator, and this distinction matters directly whenever a matrix is being interpreted, since the same array of numbers could represent either kind of object depending on context.