Quadratic form

Definition

Let $F$ be a field and let $V$ be a finite-dimensional vector space over $F$. A quadratic form on $V$ is a map $Q \colon V \to F$ such that:

1. $Q(av) = a^2 Q(v)$ for all $a \in F$ and $v \in V$.
2. The map $(u, v) \mapsto Q(u + v) - Q(u) - Q(v)$ is a bilinear form on $V$.

Remarks

On $F^n$, the quadratic forms are the maps $Q(x) = \sum_{i,j} a_{ij} x_i x_j$, that is, homogeneous polynomials of degree $2$. In matrix notation, $Q(x) = x^T A x$ for an $n \times n$ matrix $A$. If the characteristic of $F$ is not $2$, the matrix $A$ can be chosen symmetric, and then it is unique. For example, $Q(x, y) = x^2 + 3xy + 2y^2$ is a quadratic form on $\R^2$.

Explanations

L1 human

A quadratic form is a polynomial in which every term has degree two, such as $x^2 - 4xy + y^2$. It is the simplest kind of function after linear functions. The squared length $x_1^2 + \cdots + x_n^2$ is the main example.