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(Hermitian form and Hermitian space)
Let $V$ be a real vector space equipped with a complex structure $J\colon V \to V$. Then a Hermitian form on $V$ is
a complex-valued real-bilinear form
such that this is symmetric sesquilinear, in that:
$h$ is complex-linear in the first argument;
$h(w,v) = \left(h(v,w) \right)^\ast$ for all $v,w \in V$
where $(-)^\ast$ denotes complex conjugation.
A Hermitian form is positive definite (often assumed by default) if for all $v \in V$
$h(v,v) \geq 0$
$h(v,v) = 0 \phantom{AA} \Leftrightarrow \phantom{AA} v = 0$.
A complex vector space $(V,J)$ equipped with a (positive definite) Hermitian form $h$ is called a (positive definite) Hermitian space.
(basic properties of Hermitian forms)
Let $((V,J),h)$ 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 $g$ on $V$ appearing from (and fully determining) Hermitian forms $h$ via $h = g - i \omega$ are precisely those for which
These are called the Hermitian metrics.
The positive-definiteness of $g$ is immediate from that of $h$. The symmetry of $g$ follows from the symmetric sesquilinearity of $h$:
That $h$ is invariant under $J$ follows from its sesquilinarity
and this immediately implies the corresponding invariance of $g$ and $\omega$.
Analogously it follows that $\omega$ 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 $V$ with a linear complex structure $J$, then the following are equivalent:
$\omega \in \wedge^2 V^\ast$ is a linear Kähler structure (def. );
$g \in V \otimes V \to \mathbb{R}$ is a Hermitian metric (2)
where $\omega$ and $g$ 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.