PlanetMath: Hermitian form

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Hermitian form

(Definition)

A sesquilinear form over a complex vector space is a function $B: V \times V \longrightarrow \mathbb{C}$ with the properties:

$B({{\bf x}}+{{\bf y}},{{\bf z}}) = B({{\bf x}},{{\bf z}}) + B({{\bf y}},{{\bf z}})$
$B({{\bf x}},{{\bf y}}+{{\bf z}}) = B({{\bf x}},{{\bf y}}) + B({{\bf x}},{{\bf z}})$
$B(c{{\bf x}}, d{{\bf y}}) = c B({{\bf x}},{{\bf y}}) \overline{d}$

for all ${{\bf x}},{{\bf y}},{{\bf z}}\in V$ and $c,d \in \mathbb{C}$ .

A Hermitian form is a sesquilinear form which is also complex conjugate symmetric: $B({{\bf x}},{{\bf y}}) = \overline{B({{\bf y}},{{\bf x}})}$ .

An inner product over a complex vector space is a positive definite Hermitian form.

"Hermitian form" is owned by djao.

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Other names:

sesquilinear form, sesqui-linear form

Attachments:: sesquilinear forms over general fields (Definition) by Algeboy

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Cross-references: positive definite, inner product, symmetric, complex conjugate, properties, function, vector space, complex
There are 6 references to this entry.

This is version 4 of Hermitian form, born on 2002-02-22, modified 2002-08-03.
Object id is 2468, canonical name is HermitianForm.
Accessed 7127 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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