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\begin{document}

%-------------------------------------------------------------------


\section*{Hermitian form}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{linear_algebra}{}\paragraph*{{Linear algebra}}\label{linear_algebra}

[[!include homotopy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{genera_properties}{Genera properties}\dotfill \pageref*{genera_properties} \linebreak
\noindent\hyperlink{relation_to_khler_spaces}{Relation to Kähler spaces}\dotfill \pageref*{relation_to_khler_spaces} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{HermitianForm}\hypertarget{HermitianForm}{}
\textbf{([[Hermitian form]] and [[Hermitian space]])}

Let $V$ be a [[real vector space]] equipped with a [[complex structure]] $J\colon V \to V$. Then a \emph{[[Hermitian form]]} on $V$ is

\begin{itemize}%
\item a complex-valued real-[[bilinear form]]

\begin{displaymath}
h \;\colon\; V \otimes V \longrightarrow \mathbb{C}
\end{displaymath}


\end{itemize}
such that this is \emph{symmetric sesquilinear}, in that:

\begin{enumerate}%
\item $h$ is complex-linear in the first argument;


\item $h(w,v) = \left(h(v,w) \right)^\ast$ for all $v,w \in V$



\end{enumerate}
where $(-)^\ast$ denotes [[complex conjugation]].

A Hermitian form is \emph{positive definite} (often assumed by default) if for all $v \in V$

\begin{enumerate}%
\item $h(v,v) \geq 0$


\item $h(v,v) = 0 \phantom{AA} \Leftrightarrow \phantom{AA} v = 0$.



\end{enumerate}
A [[complex vector space]] $(V,J)$ equipped with a (positive definite) Hermitian form $h$ is called a (positive definite) \emph{[[Hermitian space]]}.

\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{genera_properties}{}\subsubsection*{{Genera properties}}\label{genera_properties}

\begin{prop}
\label{BasicPropertiesOfHermitianForms}\hypertarget{BasicPropertiesOfHermitianForms}{}
\textbf{(basic properties of [[Hermitian forms]])}

Let $((V,J),h)$ be a positive definite [[Hermitian space]] (def. \ref{HermitianForm}). Then

\begin{enumerate}%
\item the [[real part]] of the [[Hermitian form]]

\begin{displaymath}
g(-,-) \;\coloneqq\; Re(h(-,-))
\end{displaymath}
is a [[Riemannian metric]], hence a symmetric positive-definite real-[[bilinear form]]

\begin{displaymath}
g \;\colon\; V \otimes V \to \mathbb{R}
\end{displaymath}

\item the [[imaginary part]] of the [[Hermitian form]]

\begin{displaymath}
\omega(-,-) \;\coloneqq\; -Im(h(-,-))
\end{displaymath}
is a [[symplectic form]], hence a non-degenerate skew-symmetric real-[[bilinear form]]

\begin{displaymath}
\omega \;\colon\; V \wedge V \to \mathbb{R}
  \,.
\end{displaymath}


\end{enumerate}
hence

\begin{displaymath}
h = g - i \omega
  \,.
\end{displaymath}
The two components are related by

\begin{equation}
\omega(v,w)
  \;=\;
  g(J(v),w)
  \phantom{AAAAA}
  g(v,w)
  \;=\;
  \omega(v,J(v))
  \,.
\label{RelationBetweennOmegaAndgOnHermitianSpace}\end{equation}
Finally

\begin{displaymath}
h(J(-),J(-)) = h(-,-)
\end{displaymath}
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

\begin{equation}
g(J(-),J(-)) = g(-,-)
  \,.
\label{HermitianMetric}\end{equation}
These are called the \emph{[[Hermitian metrics]]}.

\end{prop}
\begin{proof}
The positive-definiteness of $g$ is immediate from that of $h$. The symmetry of $g$ follows from the symmetric sesquilinearity of $h$:

\begin{displaymath}
\begin{aligned}
    g(w,v)
    & \coloneqq
    Re(h(w,v))
    \\
    & =
    Re\left( h(v,w)^\ast\right)
    \\
    & =
    Re(h(v,w))
    \\
    & =
    g(v,w)
    \,.
  \end{aligned}
\end{displaymath}
That $h$ is invariant under $J$ follows from its sesquilinarity

\begin{displaymath}
\begin{aligned}
    h(J(v),J(w))
    &=
    i h(v,J(w))
    \\
    & =
    i (h(J(w),v))^\ast
    \\
    & =
    i (-i) (h(w,v))^\ast
    \\
    & =
    h(v,w)
  \end{aligned}
\end{displaymath}
and this immediately implies the corresponding invariance of $g$ and $\omega$.

Analogously it follows that $\omega$ is skew symmetric:

\begin{displaymath}
\begin{aligned}
    \omega(w,v)
    & \coloneqq
    -Im(h(w,v))
    \\
    & =
    -Im\left( h(v,w)^\ast\right)
    \\
    & =
    Im(h(v,w))
    \\
    & = - \omega(v,w)
    \,,
  \end{aligned}
\end{displaymath}
and the relation between the two components:

\begin{displaymath}
\begin{aligned}
    \omega(v,w)
    & =
    - Im(h(v,w))
    \\
    & =
    Re(i h(v,w))
    \\
    & =
    Re(h(J(v),w))
    \\
    & = 
    g(J(v),w)
  \end{aligned}
\end{displaymath}
as well as

\begin{displaymath}
\begin{aligned}
    g(v,w)
    & =
    Re(h(v,w)
    \\
    & =
    Im(i h(v,w))
    \\
    & =
    Im(h(J(v),w))
    \\
    & =
    Im(h(J^2(v),J(w)))
    \\
    & = 
    - Im(h(v,J(w)))
    \\
    & = \omega(v,J(w))
    \,.
  \end{aligned}
\end{displaymath}
\end{proof}
\hypertarget{relation_to_khler_spaces}{}\subsubsection*{{Relation to Kähler spaces}}\label{relation_to_khler_spaces}

\begin{prop}
\label{RelationBetweenKählerVectorSpacesAndHermitianSpaces}\hypertarget{RelationBetweenKählerVectorSpacesAndHermitianSpaces}{}
\textbf{(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:

\begin{enumerate}%
\item $\omega \in \wedge^2 V^\ast$ is a [[linear Kähler structure]] (def. \ref{KaehlerVectorSpace});


\item $g \in V \otimes V \to  \mathbb{R}$ is a [[Hermitian metric]] \eqref{HermitianMetric}



\end{enumerate}
where $\omega$ and $g$ are related by \eqref{RelationBetweennOmegaAndgOnHermitianSpace}

\begin{displaymath}
\omega(v,w)
  \;=\;
  g(J(v),w)
  \phantom{AAAAA}
  g(v,w)
  \;=\;
  \omega(v,J(v))
  \,.
\end{displaymath}
\end{prop}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[bilinear form]], [[quadratic form]], [[sesquilinear form]]


\item [[symplectic form]], [[Kähler form]]


\item [[hermitian matrix]]


\item [[Hermitian manifold]]


\item [[quadratic form]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[C. T. C. Wall]], \emph{On the axiomatic foundations of the theory of Hermitian forms}, Proc. Camb. Phil. Soc. (1970), 67, 243


\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Sesquilinear_form#Hermitian_form}{Hermitian form}}



\end{itemize}
[[!redirects Hermitian forms]]

[[!redirects hermitian form]] [[!redirects hermitian forms]]

[[!redirects Hermitian space]] [[!redirects Hermitian spaces]]

[[!redirects hermitian space]] [[!redirects hermitian spaces]]

[[!redirects Hermitian metric]] [[!redirects Hermitian metrics]]

[[!redirects hermitian metric]] [[!redirects hermitian metrics]]



\end{document}