15.5 Tensor Symmetric Bilinear Form Structure

Tensor Symmetric Bilinear Form Structure is the identification of a rank-2 symmetric tensor with a bilinear form on a vector space that returns the same value regardless of the order in which its two vector arguments are supplied. Given a symmetric tensor T with components T_{ij} satisfying T_{ij} = T_{ji}, the associated bilinear form B assigns to any pair of vectors u and v the scalar obtained by contracting T against u and v in both index slots, and this scalar is unchanged when u and v are exchanged, precisely because the underlying component array is symmetric.

This structure links two viewpoints on the same object: the purely algebraic viewpoint, in which a symmetric tensor is an array of numbers obeying an index equality constraint, and the functional viewpoint, in which the same object is a map from pairs of vectors to scalars obeying a symmetry property under argument exchange. The correspondence between these viewpoints is exact and reversible, so every symmetric bilinear form determines a unique symmetric rank-2 tensor and every symmetric rank-2 tensor determines a unique symmetric bilinear form.

The Bilinear Form Induced by a Symmetric Tensor

Contraction Against Two Vectors

For a symmetric tensor T with components T_{ij} on a d-dimensional vector space, the induced bilinear form is defined by:

where u^i and v^j denote the components of u and v in a chosen basis.

Symmetry of the Induced Form

Exchanging u and v in the definition and relabeling the summation indices i and j shows that B(u, v) equals B(v, u) exactly when T_{ij} equals T_{ji}, so the symmetry of the bilinear form is a direct restatement of the symmetric component equality constraint on the tensor.

The Symmetric Tensor Induced by a Bilinear Form

Extracting Components From the Form

Conversely, given a symmetric bilinear form B on a vector space with a chosen basis e_1 through e_d, a symmetric tensor is recovered by evaluating the form on pairs of basis vectors:

Consistency of the Two Constructions

Applying the tensor-to-form construction followed by the form-to-tensor construction returns the original tensor, and applying the constructions in the opposite order returns the original form, so the correspondence between symmetric rank-2 tensors and symmetric bilinear forms is a bijection that respects both directions.

Matrix Representation

Symmetric Matrix as Coordinate Realization

In a fixed basis, the component array T_{ij} can be arranged into a square matrix, and the equality constraint T_{ij} = T_{ji} is exactly the statement that this matrix equals its own transpose. Evaluating the bilinear form on vectors u and v then coincides with the matrix expression u^T T v, where u and v are written as column vectors of coordinates.

Basis Change Behavior

Under a change of basis given by an invertible matrix A, the coordinate matrix of the induced bilinear form transforms as A^T T A, and this transformed matrix remains symmetric whenever the original matrix is symmetric, since the transpose of A^T T A equals A^T T^T A, which equals A^T T A when T is symmetric.

Structural Consequences

Diagonalizability Over the Reals

When the underlying field is the real numbers and the vector space carries a standard inner product, the matrix of a symmetric bilinear form can always be diagonalized by an orthogonal change of basis, reducing the bilinear form to a sum of scaled squares of coordinates. This property depends specifically on the symmetric equality constraint and does not hold for a general, non-symmetric bilinear form.

Relation to Quadratic Forms

Every symmetric bilinear form determines an associated quadratic form by setting the two arguments equal, Q(v) = B(v, v), and in characteristic different from two this quadratic form determines the original symmetric bilinear form uniquely through a polarization identity, showing that the symmetric bilinear form structure and the quadratic form built from a symmetric tensor carry exactly the same information.