PlanetMath: positive definite form

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positive definite form

(Definition)

A bilinear form on a real or complex vector space is positive definite if for all nonzero vectors $x \in V$ . On the other hand, if for all nonzero vectors $x \in V$ , then we say is negative definite. If $B(x,x) \ge 0$ for all vectors $x \in V$ , then we say is nonnegative definite. Likewise, if $B(x,x) \le 0$ for all vectors $x \in V$ , then we say is nonpositive definite.

A form which is neither positive definite nor negative definite is called indefinite.

"positive definite form" is owned by djao.

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Other names:

positive definite, negative definite form, negative definite, indefinite form, indefinite, nonnegative definite, nonpositive definite

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Cross-references: vectors, vector space, complex, real, bilinear form
There are 18 references to this entry.

This is version 2 of positive definite form, born on 2002-02-22, modified 2004-09-20.
Object id is 2472, canonical name is PositiveDefiniteForm.
Accessed 16201 times total.

Classification:

AMS MSC:	11E39 (Number theory :: Forms and linear algebraic groups :: Bilinear and Hermitian forms)
	15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
	47A07 (Operator theory :: General theory of linear operators :: Forms )

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