15.15.2 Tensor Symmetric Matrix Bilinear Form Relation

Tensor Symmetric Matrix Bilinear Form Relation is the correspondence identifying every symmetric order-two tensor with a symmetric bilinear form on the underlying vector space, and identifying the algebraic operations, invariants, and classification results of one theory directly with those of the other.

The Correspondence

From Symmetric Tensor to Bilinear Form

A symmetric tensor T of order two over a vector space V assigns, through the standard pairing of tensors with multilinear functionals, a scalar to every pair of vectors x and y in V:

and this assignment B is bilinear, meaning linear separately in x and in y. The Component Constraint of tensor symmetry, requiring the component indexed by (i, j) to equal the component indexed by (j, i), translates exactly into the defining property of a symmetric bilinear form:

for all vectors x and y.

From Bilinear Form to Symmetric Tensor

Conversely, any symmetric bilinear form B on V determines a symmetric order-two tensor by recording its values on pairs of basis vectors as the tensor's components, and the symmetry of B forces the resulting component array to satisfy the Component Constraint. This assignment is inverse to the one above, so that symmetric order-two tensors and symmetric bilinear forms on V are in exact, structure-preserving bijection.

Compatibility of Operations

Change of Basis

Under a change of basis given by an invertible matrix P, the components of the tensor transform by the congruence rule,

and this is exactly the transformation rule that a symmetric bilinear form's matrix undergoes when the coordinates of the underlying vectors are changed by P. The two theories therefore share not only their objects but the entire group of coordinate changes acting on those objects.

Associated Quadratic Form

Every symmetric bilinear form B determines, and is determined by (in characteristic different from two), an associated quadratic form Q given by evaluating B on equal arguments:

so that the pure power form of x, in the tensor picture, corresponds precisely to evaluating the quadratic form at x. This is the order-two instance of the general correspondence between symmetric tensors and homogeneous polynomials, restricted here to degree two.

Invariant-Theoretic Consequences

Rank as a Shared Invariant

The rank of the matrix representing T, invariant under the congruence transformations induced by basis change, is simultaneously the linear-algebraic rank of the tensor and the rank of the associated bilinear form, defined as the dimension of V minus the dimension of the radical (the subspace of vectors x for which B(x, y) vanishes for every y). This shared rank is also, by the equality of symmetric and ordinary rank established under the Rank Relation for order two, the symmetric tensor rank of T.

Signature and Classification

Over the real numbers, the congruence classification of symmetric bilinear forms by Sylvester's law of inertia, which sorts forms by the number of positive, negative, and zero eigenvalues (the signature), applies verbatim to symmetric order-two tensors, giving a complete list of canonical representatives, one for each possible signature, up to change of basis. Over an algebraically closed field, the classification collapses further to rank alone, since every nonzero scalar is a square, and this is mirrored exactly in the classification of order-two pure-power-form decompositions.

Definiteness and Positivity

A symmetric bilinear form is called positive definite when its associated quadratic form takes only positive values on nonzero vectors; translated into tensor language, this is the condition that the symmetric tensor, viewed as a matrix, has every eigenvalue positive. Positive definiteness of the tensor is precisely what permits its interpretation as a genuine inner product, which underlies the use of symmetric, positive-definite order-two tensors as metric tensors in differential geometry and as covariance tensors in probability and statistics.

Extension Beyond Order Two

Multilinear Forms as the General Pattern

The Bilinear Form Relation is the order-two case of a general correspondence between symmetric tensors of order d and symmetric d-linear forms, functions of d vector arguments that are linear in each argument separately and invariant under permuting the arguments. Evaluating a symmetric d-linear form on d copies of the same vector recovers the associated degree-d homogeneous polynomial, generalizing the quadratic form obtained in the order-two case.

Why Order Two Remains Distinguished

Because bilinear forms reduce, via diagonalization, to a sum of independent one-variable pieces (a consequence of the spectral theorem discussed under the Component Constraint), the order-two Bilinear Form Relation yields a complete and simple classification that has no direct counterpart for multilinear forms of order three or higher, where diagonal-like simultaneous decompositions are governed instead by the more intricate theory of the Rank Relation, Term Set uniqueness, and Reconstruction.