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(Hermitian form and Hermitian space)
Let be a real vector space equipped with a complex structure . Then a Hermitian form on is
a complex-valued real-bilinear form
such that this is symmetric sesquilinear, in that:
is complex-linear in the first argument;
for all
where denotes complex conjugation.
A Hermitian form is positive definite (often assumed by default) if for all
.
A complex vector space equipped with a (positive definite) Hermitian form is called a (positive definite) Hermitian space.
(basic properties of Hermitian forms)
Let be a positive definite Hermitian space (def. ). Then
the real part of the Hermitian form
is a Riemannian metric, hence a symmetric positive-definite real-bilinear form
the imaginary part of the Hermitian form
is a symplectic form, hence a non-degenerate skew-symmetric real-bilinear form
hence
The two components are related by
Finally
and so the Riemannian metrics on appearing from (and fully determining) Hermitian forms via are precisely those for which
These are called the Hermitian metrics.
The positive-definiteness of is immediate from that of . The symmetry of follows from the symmetric sesquilinearity of :
That is invariant under follows from its sesquilinarity
and this immediately implies the corresponding invariance of and .
Analogously it follows that is skew symmetric:
and the relation between the two components:
as well as
(relation between Kähler vector spaces and Hermitian spaces)
Given a real vector space with a linear complex structure , then the following are equivalent:
is a linear Kähler structure (def. );
is a Hermitian metric (2)
where and are related by (1)
C. T. C. Wall, On the axiomatic foundations of the theory of Hermitian forms, Proc. Camb. Phil. Soc. (1970), 67, 243
Wikipedia, Hermitian form
Last revised on June 11, 2022 at 07:09:19. See the history of this page for a list of all contributions to it.