Linear and multilinear algebra; matrix theory Definition

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:

$Q(av) = a^2 Q(v)$ for all $a \in F$ and $v \in V$ .
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 $\R^2$ .

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Vector space definition
Bilinear form definition
Quadratic form definition

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Tag	`0HZNE5QZDJ`
URI	`m2://0HZNE5QZDJ`
Kind	definition
Writing	W1 stated
MSC	15A63, 11E04
Defines	Quadratic form
Uses	Vector space, Bilinear form

Quadratic form

Definition

Remarks

Reading path

Where this leads

Further reading

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