Diagonalisation of a quadratic form

Builds on Quadratic form Orthogonal family for a bilinear form

Statement

Let $F$ be a field of characteristic not $2$ , and let $V$ be a vector space over $F$ of finite dimension $n$ . Let $Q$ be a quadratic form on $V$ . Then $V$ has a basis $(e_1, \dots, e_n)$ and there are $a_1, \dots, a_n \in F$ such that, for all $x_1, \dots, x_n \in F$ , $Q(x_1 e_1 + \cdots + x_n e_n) = a_1 x_1^2 + a_2 x_2^2 + \cdots + a_n x_n^2.$

Remarks

The basis is orthogonal for the associated symmetric bilinear form, and $a_i = Q(e_i)$ . In matrix terms, each symmetric matrix $A$ over $F$ has an invertible $P$ such that $P^T A P$ is diagonal. The numbers $a_i$ are not unique; for example, each $a_i$ can be multiplied by a nonzero square. Over $\R$ , the orthogonal diagonalisation of the principal axis theorem is a special case.

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Vector space definition
Bilinear form definition
Orthogonal family for a bilinear form definition
Quadratic form definition
Diagonalisation of a quadratic form statement

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Classification of complex quadratic forms

Tag	`0Z3FEWZ8HA`
URI	`m2://0Z3FEWZ8HA`
Kind	statement, theorem
Writing	W0 stub
MSC	11E04, 15A63
Status	This copy of the site has no build data. Run `m2 status 0Z3FEWZ8HA`.
Uses	Quadratic form, Orthogonal family for a bilinear form

Diagonalisation of a quadratic form

Statement

Remarks

Reading path

Where this leads

Further reading

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