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algebra > linear algebra > vector space > Hermitian space

Preferred term

Hermitian space

Definition(s)

A complex Hermitian form (also called a symmetric sesquilinear form), is a sesquilinear form ${\\displaystyle h:V\ imes V\ o \\mathbb {C} }$ such that
${\\displaystyle h(w,z)={\\overline {h(z,w)}}.}$
The standard Hermitian form on ${\\displaystyle \\mathbb {C} ^{n}}$ is given (again, using the "physics" convention of linearity in the second and conjugate linearity in the first variable) by
${\\displaystyle \\langle w,z\ angle =\\sum _{i=1}^{n}{\\overline {w}}_{i}z_{i}.}$
More generally, the inner product on any complex Hilbert space is a Hermitian form.
A minus sign is introduced in the Hermitian form ${\\displaystyle ww^{*}-zz^{*}}$ to define the group SU(1,1).
A vector space with a Hermitian form ${\\displaystyle (V,h)}$ is called a Hermitian space.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Sesquilinear_form#Hermitian_form)

Broader concept(s)

vector space

In other languages

espace hermitien

French

URI

http://data.loterre.fr/ark:/67375/PSR-CF9WKGJL-W

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