16.8.3 Tensor Alternating Bilinear Matrix Representation

Tensor Alternating Bilinear Matrix Representation is the explicit procedure for converting an alternating bilinear form into a skew-symmetric matrix relative to a chosen basis, together with the rules governing how that matrix behaves under evaluation, basis change, and composition with other linear data.

Constructing the Matrix

Entry-by-Entry Definition

Given a basis e₁, ..., eₙ for V, the matrix representation M of an alternating bilinear form B is built by evaluating B on every pair of basis vectors:

Because B is alternating, this matrix automatically satisfies M_{ii} = 0 and M_{ji} = −M_{ij}, so it is skew-symmetric by construction rather than by an additional imposed condition.

Evaluating B via the Matrix

Once M is built, B can be evaluated on arbitrary vectors u = Σuᵢeᵢ and v = Σvⱼeⱼ using ordinary matrix multiplication:

reducing the abstract multilinear evaluation to a finite double sum over matrix entries.

Behavior Under Basis Change

The Congruence Transformation Law

If the basis changes according to an invertible matrix A (so new basis vectors are given by columns of A in terms of the old basis), the matrix representation transforms by congruence, not by the ordinary similarity transformation used for linear operators:

This distinction matters: bilinear forms transform by congruence because they take two vector arguments, whereas linear operators (which take one vector argument and return one vector) transform by similarity A⁻¹MA.

Preservation of Skew-Symmetry Under Congruence

The congruence transformation automatically preserves skew-symmetry: (M')ᵀ = AᵀMᵀA = Aᵀ(−M)A = −M', confirming that the matrix representation of an alternating bilinear form remains a valid skew-symmetric matrix in every basis, not merely the original one.

Rank and Its Matrix-Level Meaning

Rank as an Intrinsic Invariant

The rank of the matrix M is a basis-independent invariant of B itself, since congruence transformations preserve matrix rank; this rank determines the size of the largest nondegenerate alternating "block" contained within B and, by a standard structural fact, is always even.

Kernel of the Matrix Representation

Vectors v satisfying Mv = 0 — equivalently, B(u,v) = 0 for every u — form the radical (kernel) of B; the matrix representation makes this subspace directly computable via ordinary linear algebra, namely finding the null space of M.

Worked Matrix Construction

A Rank-4 Example

For n = 4 with basis e₁,...,e₄, suppose B(e₁,e₂) = 3, B(e₁,e₃) = 0, B(e₁,e₄) = 0, B(e₂,e₃) = 0, B(e₂,e₄) = 0, B(e₃,e₄) = 5; the matrix representation is then fully determined:

with rank exactly 4, matching its block-diagonal symplectic canonical form.

Matrix Representation and the Wedge Product

Reading Wedge Basis Elements Off the Matrix

Each off-diagonal entry M_{ij} for i < j corresponds exactly to the coefficient of the basis wedge element eᵢ* ∧ eⱼ* when B is expanded in the standard basis of Λ²(V), giving a direct dictionary between the matrix's upper-triangular entries and the wedge-product expansion coefficients.

Diagram of the Representation Pipeline