15.16.4 Tensor Quadratic Form Polarization Relation

Tensor Quadratic Form Polarization Relation is the family of algebraic identities that convert between a quadratic form's values on single vectors and its underlying symmetric bilinear tensor's values on pairs of vectors, together with the variants of this identity, its interpretation as a directional derivative, and its generalization to symmetric tensors of higher order.

The Core Identity

Standard Polarization Formula

Given a quadratic form Q arising from a symmetric bilinear tensor T via Q(x) equals T(x, x), the Polarization Relation recovers T at an arbitrary pair of vectors from three evaluations of Q:

This identity follows directly from expanding Q(x + y) as T(x + y, x + y) and using bilinearity and symmetry of T to collect the cross terms, which appear exactly twice because T(x, y) equals T(y, x).

The Difference-of-Squares Variant

An equivalent form of the Polarization Relation, sometimes more convenient numerically since it avoids subtracting two separately computed diagonal values, is obtained from the parallelogram-type expansion,

which follows from expanding both Q(x + y) and Q(x - y) and observing that the pure terms in x and in y cancel upon subtraction, leaving only the cross term, now appearing with coefficient four rather than two.

Interpretive Content

Polarization as a Directional Derivative

The Polarization Relation admits a calculus interpretation: because Q is homogeneous of degree two, the quantity T(x, y) obtained by polarization is, up to the same constant factor of one half, exactly the derivative of Q at the point x in the direction y, computed via the limit definition of the directional derivative applied to the smooth function Q. This is the reason the associated tensor T coincides, as noted under the Tensor Quadratic Form Component Expression, with (half of) the Hessian of Q, since the Hessian is built from second derivatives and T is itself already a first-derivative-like polarization of Q in one argument, applied to the linear gradient of Q in the other.

Polarization as an Averaging Procedure

More conceptually, polarization can be understood as recovering the "off-diagonal information" of a symmetric bilinear tensor that is compressed, by the act of setting both arguments equal, into a single-variable function; the Polarization Relation is the precise bookkeeping needed to undo that compression using only a finite number of evaluations of the compressed function.

Dependence on the Characteristic of the Field

The Role of the Factor One Half

Both variants of the Polarization Relation require dividing by two (or by four), and this division is only meaningful when the underlying field does not have characteristic two. This is exactly the condition identified under the Tensor Quadratic Form Symmetric Bilinear Source as the requirement for the Symmetric Bilinear Source of a quadratic form to exist and be unique, confirming that the Polarization Relation is the explicit formula realizing that existence and uniqueness claim.

Formal Polarization Without Division

In settings where division by two is unavailable or undesirable, an unnormalized version of the identity,

still holds and remains useful, though it determines T only up to the ambiguity inherent in dividing by two, which is precisely the obstruction handled separately by the specialized theory of quadratic forms in characteristic two.

Generalization to Higher Order

Polarization of Cubic and Higher Forms

The same underlying idea extends to symmetric tensors of order d greater than two and their associated degree-d homogeneous polynomials: evaluating the polynomial at sums of several vectors with various signs, and taking an appropriate combination of the results with binomial-type coefficients, recovers the full d-linear symmetric tensor from its diagonal restriction. This general polarization procedure is the higher-order counterpart of the Polarization Relation described here, and it is the reason apolarity-based methods for Tensor Symmetric Decomposition Reconstruction can operate directly on the polynomial form of a tensor without loss of information, provided the characteristic of the field exceeds the order d.

Consistency With the Order-Two Case

Setting d equal to two in the general polarization procedure reduces exactly to the identity presented at the start of this discussion, confirming that the Tensor Quadratic Form Polarization Relation is not a special, isolated trick but the smallest instance of a single, uniform mechanism operating throughout symmetric tensor theory.