15.16.1 Tensor Quadratic Form Symmetric Bilinear Source

Tensor Quadratic Form Symmetric Bilinear Source is the identification of the symmetric bilinear form (equivalently, the symmetric order-two tensor) that lies behind every quadratic form, together with the precise account of why this underlying source exists, when it is unique, and how it fails to exist or fails to be unique when the characteristic of the base field obstructs the usual argument.

The Source Object Behind a Quadratic Form

Quadratic Forms Do Not Arise in a Vacuum

A quadratic form Q on a vector space V is, on its own, simply a function satisfying homogeneity of degree two and a parallelogram-type expansion law. The Tensor Quadratic Form Relation shows that every such Q arises from evaluating some symmetric bilinear tensor T on a repeated argument. The Symmetric Bilinear Source is precisely this tensor T: the underlying, generally hidden, two-argument object whose diagonal restriction produces the one-argument function Q that is directly observed or specified in practice.

Existence of the Source

The existence of a symmetric bilinear source for any given quadratic form Q is guaranteed, over a field whose characteristic is not two, by the polarization identity,

which constructs T explicitly from Q alone, without any additional data, by using Q evaluated at three points: x, y, and their sum.

Verifying the Constructed Source

Bilinearity of the Constructed Object

The function T produced by the polarization identity can be checked directly to be linear in each of its two arguments separately, using the defining homogeneity and expansion properties of Q, and this bilinearity is what qualifies T as a genuine order-two tensor rather than merely an auxiliary formula.

Symmetry of the Constructed Object

The formula for T is manifestly unchanged under swapping x and y, since Q(x + y) equals Q(y + x) and the remaining terms are symmetric by inspection; this immediate symmetry is what identifies the constructed T as lying specifically in the space of symmetric tensors, satisfying the Component Constraint, rather than in the larger space of general bilinear forms.

Recovery of the Original Form

Substituting y equal to x into the polarization formula returns Q(x) exactly, up to a routine simplification of the right-hand side, confirming that the constructed T is indeed a Symmetric Bilinear Source for Q in the precise sense required: evaluating T on the diagonal reproduces Q.

Uniqueness of the Source

Uniqueness Away from Characteristic Two

Because the polarization formula is the only way to build a symmetric bilinear functional from Q using solely linear combinations of values of Q, and because any two symmetric bilinear forms restricting to the same quadratic form must have a vanishing antisymmetric-like difference forced to zero by symmetry, the Symmetric Bilinear Source of a given quadratic form is unique whenever the characteristic of the field is not two. This uniqueness is what allows quadratic forms and symmetric order-two tensors to be treated, in ordinary settings, as fully interchangeable objects.

Breakdown in Characteristic Two

In characteristic two, the coefficient one half used in the polarization identity is not defined, and the correspondence between quadratic forms and symmetric bilinear tensors genuinely breaks down: distinct quadratic forms can share the same associated symmetric bilinear form obtained by a characteristic-two-adapted construction, and the theory instead requires the separate, more refined machinery of quadratic forms over fields of characteristic two, in which the quadratic form is treated as primary data and the bilinear form it determines, via a modified polarization formula without the factor of one half, carries strictly less information than Q itself.

Structural Role of the Source

The Source as the Object Carrying Tensorial Structure

While Q is a scalar-valued function of one vector, its Symmetric Bilinear Source T is a genuine element of the tensor product space, and it is T, not Q directly, that transforms under change of basis by the congruence rule, that possesses a rank and a definiteness classification, and that admits a symmetric decomposition into pure power forms. Every structural and computational tool associated with quadratic forms, including diagonalization and the spectral theorem in the Matrix Case, is in fact a tool applied to the Symmetric Bilinear Source rather than to the quadratic form as a bare function.

The Source as the Bridge to General Symmetric Tensor Theory

Because the Symmetric Bilinear Source is an order-two symmetric tensor, it inherits every general notion developed for symmetric tensors of arbitrary order, including tensor rank and the Rank Relation comparing it to symmetric rank, even though at order two these notions coincide. Identifying the Symmetric Bilinear Source explicitly is therefore the step that connects the elementary, function-based description of a quadratic form to the full apparatus of symmetric tensor algebra, decomposition, and classification developed throughout the broader theory.