Definition:Quadratic Form (Linear Algebra)

Definition

Let $\mathbb K$ be a field of characteristic $\Char {\mathbb K} \ne 2$.

Let $V$ be a vector space over $\mathbb K$.

A quadratic form on $V$ is a mapping $q : V \mapsto \mathbb K$ such that:

$\forall v \in V : \forall \kappa \in \mathbb K : \map q {\kappa v} = \kappa^2 \map q v$

$b: V \times V \to \mathbb K: \tuple {v, w} \mapsto \map q {v + w} - \map q v - \map q w$ is a bilinear form

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Also defined as

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A quadratic form is a homogeneous polynomial of degree $2$.

Example:

$x^2 + 2 x y - 3 y^2 + 4 x z$

is a quadratic form in the variables $x$, $y$ and $z$.

Also see

• Definition:Bilinear Form (Linear Algebra)
• Definition:Associated Bilinear Form

• Results about quadratic forms in the context of linear algebra can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): quadratic (quadric)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): quadratic
• 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $7$: Patterns in Numbers: Gauss
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): quadratic form