6.21.3 Tensor Matrix Bilinear Form Context

Tensor Matrix Bilinear Form Context is the reading of a second-order covariant tensor as a bilinear form, that is, as a matrix B that takes two vectors and produces a scalar via B(v, w) = vᵀBw, rather than as a linear map that transforms one vector into another. This context sits alongside the linear-map context as the second major way a second-order tensor's matrix is interpreted in practice, and the two interpretations, while built from the same array of numbers, correspond to genuinely different mathematical objects with different transformation laws.

The Bilinear Form Reading of a Matrix

From Tensor to Two-Slot Function

A purely covariant second-order tensor B ∈ V* ⊗ V* is, by definition, a bilinear functional V × V → F. Once a basis {eᵢ} of V is fixed, B is completely determined by its values Bᵢⱼ = B(eᵢ, eⱼ) on pairs of basis vectors, and for general vectors v = Σvⁱeᵢ, w = Σwʲeⱼ, bilinearity gives

which in matrix notation is the familiar quadratic-form-style expression vᵀBw, with both v and w entering as column vectors and B sandwiched between them.

Two Inputs, Not One Input and One Output

The essential structural difference from the linear-map context is arity: a bilinear form consumes two vectors from the same space V and returns a scalar, while a linear map consumes one vector and returns a vector in a possibly different space W. The tensor B in V* ⊗ V* and the tensor T in V* ⊗ W may have numerically identical matrices in some basis, yet represent conceptually distinct objects because of this difference in how the matrix is meant to be used.

Why the Transformation Law Differs

Congruence Instead of Similarity

Because both indices of Bᵢⱼ are covariant, a change of basis with transition matrix A transforms the bilinear form's matrix by congruence,

which is different from the similarity transformation T′ = A⁻¹TA appropriate to a (1,1) mixed tensor representing a linear map from V to itself. Confusing the two transformation rules is a common error precisely because both start from a matrix and a change-of-basis matrix A, but the correct rule depends on the variance type, not on the numerical values of the matrix.

Symmetric and Antisymmetric Bilinear Forms

Because Bᵢⱼ carries two indices of the same variance, the symmetry property Bᵢⱼ = Bⱼᵢ (or antisymmetry Bᵢⱼ = −Bⱼᵢ) is preserved by the congruence transformation and is therefore a basis-independent property of the bilinear form; this is the reason a symmetric bilinear form (such as an inner product) remains symmetric in every basis, while a general (1,1) linear map has no such invariant symmetry notion attached to it.

Diagram Contrasting the Two Contexts

Special Bilinear Forms Arising From This Context

The Metric Tensor as a Bilinear Form

The most important instance of the tensor matrix bilinear form context is the metric tensor g ∈ V* ⊗ V*, a symmetric bilinear form used to define lengths and angles via g(v, v) and g(v, w). Its matrix representation, the Gram matrix, is exactly a Bᵢⱼ array subject to the congruence transformation law described above, and it is this bilinear-form character — not a linear-map character — that gives the metric its role in defining an inner product structure on V.

Quadratic Forms as a Restriction

Restricting a symmetric bilinear form to the diagonal, Q(v) = B(v, v), produces the associated quadratic form, whose matrix is the same Bᵢⱼ and whose classification (positive definite, indefinite, degenerate) is read off from basis-independent invariants of that matrix, such as the signature of its eigenvalues, obtained precisely because those eigenvalues are computed with respect to the congruence transformation law rather than the similarity law.