Linear and multilinear algebra; matrix theory Definition

Polar form of a quadratic form

Builds on Quadratic form Symmetric bilinear form

Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$ , and let $Q$ be a quadratic form on $V$ . The polar form of $Q$ is the map $B_Q \colon V \times V \to F$ with $B_Q(u, v) = Q(u + v) - Q(u) - Q(v).$

Remarks

$B_Q$ is a symmetric bilinear form, and $B_Q(v, v) = 2 Q(v)$ for all $v$ . For $Q(x) = x^T A x$ on $F^n$ , $B_Q(x, y) = x^T (A + A^T) y$ . Many authors divide by $2$ when the characteristic is not $2$ , so that the result $B$ satisfies $B(v, v) = Q(v)$ ; see the associated form. This construction is the degree $2$ case of the polarisation of homogeneous polynomials.

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Vector space definition
Bilinear form definition
Quadratic form definition
Symmetric bilinear form definition
Polar form of a quadratic form definition

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Symmetric bilinear form associated with a quadratic form

Tag	`02STTYFBVY`
URI	`m2://02STTYFBVY`
Kind	definition
Writing	W0 stub
MSC	15A63, 11E04
Defines	Polar form of a quadratic form
Uses	Quadratic form, Symmetric bilinear form

Polar form of a quadratic form

Definition

Remarks

Reading path

Where this leads

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