Linear and multilinear algebra; matrix theory Definition

Symmetric bilinear form associated with a quadratic form

Builds on Quadratic form Polar form of a quadratic form Symmetric bilinear form

Definition

Let $F$ be a field of characteristic not $2$ , and let $V$ be a finite-dimensional vector space over $F$ . Let $Q$ be a quadratic form on $V$ . The symmetric bilinear form associated with $Q$ is $B(u, v) = \tfrac{1}{2} \bigl( Q(u + v) - Q(u) - Q(v) \bigr) \qquad (u, v \in V).$

Remarks

$B$ is a symmetric bilinear form with $B(v, v) = Q(v)$ , and it is the only one. Conversely, each symmetric bilinear form $B$ gives the quadratic form $v \mapsto B(v, v)$ ; the two constructions are inverse to each other. For $Q(x) = x^T A x$ with $A$ symmetric, $B(x, y) = x^T A y$ . It is half of the polar form of $Q$ .

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

Tag	`0TX2YH2FQS`
URI	`m2://0TX2YH2FQS`
Kind	definition
Writing	W0 stub
MSC	15A63, 11E04
Defines	Symmetric bilinear form associated with a quadratic form
Uses	Quadratic form, Polar form of a quadratic form, Symmetric bilinear form

Symmetric bilinear form associated with a quadratic form

Definition

Remarks

Reading path

Further reading

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