15.16 Tensor Quadratic Form Relation

Tensor Quadratic Form Relation is the general principle, underlying the entire theory of symmetric tensors, that identifies a symmetric tensor of order two with a quadratic form on the underlying vector space, and that positions this identification as the foundational, order-two instance of the broader correspondence between symmetric tensors of any order and homogeneous polynomials of that same order.

The Identification in General Terms

Symmetric Tensors as Polynomial Functions

A symmetric tensor of order d on a vector space V can always be evaluated on a single vector argument, repeated d times, to produce a scalar; doing so for every vector in V defines a function on V that is homogeneous of degree d, and this function is precisely the homogeneous polynomial classically associated with the tensor. The Tensor Quadratic Form Relation is the specialization of this general correspondence to the case d equals two, in which the resulting homogeneous polynomial is, by definition, a quadratic form.

The Defining Formula

Concretely, if T is a symmetric tensor of order two and x is a vector, the associated quadratic form Q is given by full contraction of T against two copies of x:

using the interpretation of T as a symmetric bilinear functional established under the Bilinear Form Relation. This single formula is the entire content of the Tensor Quadratic Form Relation, and every other fact in this area is a consequence of unpacking it in coordinates or applying it to particular structures.

Recovering the Tensor from the Form

The Polarization Principle

Because a symmetric bilinear functional is completely determined by its values on the diagonal (that is, by its restriction to pairs of equal arguments), whenever the underlying field does not have characteristic two, the quadratic form Q alone suffices to recover the full tensor T through the polarization identity,

This bidirectional recoverability is what makes the identification of symmetric order-two tensors with quadratic forms a genuine equivalence of structures rather than a one-way construction, and it is the order-two case of a polarization procedure that extends, with more terms, to recover symmetric tensors of any order from their associated degree-d homogeneous polynomials.

Structural Consequences

Transfer of Invariants

Every basis-independent invariant of the tensor T, such as its rank and its definiteness class, transfers to a corresponding invariant of the quadratic form Q, and conversely: the rank of Q, meaning the number of variables genuinely occurring in Q after a suitable linear change of variables, equals the tensor rank of T, and the classification of Q by definiteness corresponds to the eigenvalue sign pattern of T. This transfer of invariants is what allows classical results about quadratic forms, developed independently in number theory and geometry, to be reinterpreted directly as results about symmetric tensors.

Compatibility with Decomposition

A symmetric decomposition of T into pure power forms corresponds, under the Tensor Quadratic Form Relation, to writing Q as a sum of squares of linear forms:

so that the symmetric rank of T equals the minimal number of squares of linear forms needed to express Q, connecting the general theory of symmetric tensor decomposition directly to the classical algebraic and number-theoretic study of sums of squares.

Position Within the Broader Theory

Relation to the Matrix Case

Once a basis is fixed, the tensor T underlying the Tensor Quadratic Form Relation is represented by a symmetric matrix, and the entire relationship becomes the concrete, coordinate-based Quadratic Form Relation studied within the Matrix Case, where diagonalization, the spectral theorem, and explicit rank and definiteness computations become available. The present, more general formulation is the coordinate-free statement of which that matrix-level treatment is the computational unpacking.

Relation to Higher-Order Symmetric Tensors

The Tensor Quadratic Form Relation is the base case, at order two, of the general association between symmetric tensors and homogeneous polynomials that underlies Tensor Symmetric Decomposition Structure at every order. Concepts that appear simple here, such as rank and decomposition into pure powers, reappear at higher orders in more intricate form, governed by the Rank Relation, the geometry of the Veronese variety, and the Alexander-Hirschowitz classification of exceptional cases, with the order-two quadratic form setting serving throughout as the base case against which the general theory is calibrated and tested.