The diagonal of a symmetric form
Given a symmetric bilinear form B, define its quadratic form by q(v) = B(v, v). In coordinates with Gram matrix A this is q(x) = xᵀ A x = Σᵢ aᵢᵢ xᵢ² + 2 Σ_{i<j} aᵢⱼ xᵢ xⱼ — a homogeneous polynomial of degree 2. Conversely any degree-2 homogeneous polynomial q(x₁, …, xₙ) is the quadratic form of a unique symmetric matrix: put aᵢᵢ = (coefficient of xᵢ²) and aᵢⱼ = aⱼᵢ = ½(coefficient of xᵢxⱼ). The ½ is why characteristic 2 will need special care.
A function q : V → K deserves to be called a quadratic form abstractly when (i) q(λv) = λ² q(v) for all scalars λ, and (ii) the map B_q(u, w) := q(u + w) − q(u) − q(w) is bilinear. Then B_q is automatically symmetric, and it is the *associated bilinear form* of q. Condition (i) is the homogeneity that makes “quadratic” literal: scaling the input by λ scales the output by λ².
Polarization: recovering B from q
Knowing q on the diagonal seems like less information than knowing B everywhere. The polarization identity says that, away from characteristic 2, it is exactly the same information. Expand q(u + w) = B(u + w, u + w) = q(u) + 2B(u, w) + q(w). Solve for the cross term: B(u, w) = ½ [q(u + w) − q(u) − q(w)]. So B and q determine each other, and we may speak of “a quadratic form” and “a symmetric bilinear form” interchangeably whenever 2 is invertible in K.
Polarization, worked on R^2. Form: q(x1, x2) = 3*x1^2 + 4*x1*x2 + x2^2. Gram matrix (halve the off-diagonal coefficient): A = [3, 2; 2, 1]. Check: x^T A x = 3 x1^2 + 4 x1 x2 + x2^2. Good. Recover B by polarization with u = (1,0), w = (0,1): q(u) = q(1,0) = 3 q(w) = q(0,1) = 1 q(u + w) = q(1,1) = 3 + 4 + 1 = 8 B(u, w) = (1/2)(8 - 3 - 1) = 2 = A[1,2]. Consistent. Why char 2 breaks: there 1/2 does not exist, and over F_2 x1^2 + x2^2 = (x1 + x2)^2 has associated B = 0 (it is ALTERNATING), so q carries strictly more information than its B. Quadratic forms and symmetric bilinear forms part ways in characteristic 2.
Equivalence of quadratic forms
Two quadratic forms q, q' on V are equivalent (or isometric) if there is an invertible linear change of variables carrying one into the other — exactly when their matrices are congruent. A diagonal form a₁x₁² + ⋯ + aₙxₙ² is abbreviated ⟨a₁, …, aₙ⟩. Scaling a variable xᵢ ↦ c xᵢ multiplies the i-th coefficient by c², so over any field the diagonal entries only matter up to nonzero squares: ⟨a⟩ ≅ ⟨ac²⟩. That square-class bookkeeping is the seed of everything in Guides 4 and 5.