15.5.4 Tensor Symmetric Bilinear Quadratic Relation

Tensor Symmetric Bilinear Quadratic Relation is the precise correspondence linking a symmetric bilinear form built from a symmetric tensor to the quadratic form obtained by evaluating that bilinear form on a single repeated vector argument. Given a symmetric bilinear form B, the associated quadratic form Q is defined by Q(v) = B(v, v), and the relation states not only how Q is produced from B but also how B can be recovered from Q alone, through a polarization identity that expresses the bilinear form entirely in terms of quadratic form values.

This relation depends essentially on the symmetry of B: for a general, non-symmetric bilinear form, only the symmetric part contributes to Q(v) = B(v, v), because the antisymmetric part of any bilinear form vanishes identically when its two arguments coincide. As a result, the quadratic form Q never carries information about the antisymmetric part of B, and the bilinear-quadratic relation is naturally a statement about symmetric bilinear forms specifically, since these are exactly the forms fully recoverable from their associated quadratic form.

From Bilinear Form to Quadratic Form

Diagonal Evaluation

Setting the second argument of a symmetric bilinear form equal to the first produces the quadratic form:

which is a homogeneous degree-two polynomial in the coordinates of v, with coefficients drawn directly from the components of the symmetric tensor T.

Homogeneity of the Quadratic Form

Scaling the vector v by a factor c scales the quadratic form by c squared, since both occurrences of v in B(v, v) contribute a factor of c: Q(cv) = c^2 Q(v). This degree-two homogeneity is the defining characteristic that distinguishes a quadratic form from a general bilinear evaluation.

From Quadratic Form to Bilinear Form

The Polarization Identity

Given only the quadratic form Q, the original symmetric bilinear form B can be recovered by the polarization identity:

valid whenever division by two is meaningful in the underlying field, which is the case for the real and complex numbers among others.

Verification of the Identity

Expanding Q(u + v) as B(u + v, u + v) and using bilinearity gives B(u, u) + B(u, v) + B(v, u) + B(v, v), which equals Q(u) + Q(v) + 2B(u, v) once the symmetry relation B(u, v) = B(v, u) is used to combine the two cross terms; solving this expansion for B(u, v) reproduces the polarization identity exactly.

Necessity of Symmetry in the Relation

Loss of Antisymmetric Information

If B is decomposed into a symmetric part and an antisymmetric part, only the symmetric part survives in Q(v) = B(v, v), because the antisymmetric part satisfies B_anti(v, v) = 0 for every v, a direct consequence of B_anti(v, v) = -B_anti(v, v) forcing the value to vanish. The polarization identity therefore reconstructs only the symmetric part of any original bilinear form, never the antisymmetric part.

The Relation as an Equivalence

Because of this, the bilinear-quadratic relation establishes a one-to-one correspondence specifically between symmetric bilinear forms and their quadratic forms: distinct symmetric bilinear forms always produce distinct quadratic forms, and every quadratic form arising from a bilinear evaluation arises from exactly one symmetric bilinear form.

Component-Level Expression

Diagonal and Off-Diagonal Contributions

Writing the quadratic form sum explicitly separates diagonal terms, where i equals j, from off-diagonal terms, where i differs from j:

where the factor of two on the off-diagonal sum arises because each unordered pair {i, j} with i different from j contributes twice in the original double sum, once as T_{ij} v^i v^j and once as T_{ji} v^j v^i, and these two contributions are equal by the equality constraint.

Consistency With Independent Component Counting

This expression shows directly that the independent components of the symmetric tensor, the diagonal entries T_{ii} and the off-diagonal representatives T_{ij} with i less than j, are exactly the coefficients needed to write down the quadratic form, confirming that the quadratic form and the independent selection of the symmetric tensor carry equivalent information.