sesquilinear forms over general fieldsSesquilinearFormsOverGeneralFields2006-06-09 16:16:592006-06-16 19:24:15DefinitionHermitian formsesquilinear formHermitian formbilinear formHermiteansesquilinear formHermitian form\usepackage{latexsym}
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Let $V$ be a vector space over a field $k$. $k$ may be of any characteristic.
\section{Sesquilinear Forms}
\begin{defn}
A function $b:V\times V\rightarrow k$ is sesquilinear if it satisfies each of
the following:
\begin{enumerate}
\item $b(v,w+u)=b(v,w)+b(v,u)$ and $b(v+u,w)=b(v,w)+b(u,w)$ for all $u,v,w\in V$;
\item For a given field automorphism $\theta$ of $k$, $b(v,lw)=l^\theta b(v,w)$ and $b(lv,w)=lb(v,w)$ for all $v,w\in V$ and $l\in k$.
\end{enumerate}
\end{defn}
\begin{remark}
It is possible to apply the field automorphism in the first variable but is more common to do so in the second variable. Also, if $\theta=1$ the form is a bilinear form.
\end{remark}
Sesquilinear forms are commonly ascribed any combination of the following properties:
\begin{itemize}
\item non-degenerate,
\item reflexive, (commonly required to define perpendicular);
\item positive definite (this condition requires the fixed field of $\theta$,
$k_0$, be an ordered field, such as the rationals $\mathbb{Q}$ or reals $\mathbb{R}$).
\end{itemize}
Non-degenerate sesquilinear and bilinear forms apply to projective geometries as dualities and polarities through the induced $\perp$ operation. (See \PMlinkname{polarity}{Polarity2}.)
\section{Hermitian Forms}
If $\theta^2=1$, it is common to exchange notation at this point and use the same notation of $\bar{l}$ for $l^\theta$ as is common for complex conjugation -- even if $k$ is not $\mathbb{C}$. Then $\bar{\bar{l}}=l$.
In this notation, Hermitian forms may be defined by the property
\[b(v,w)=\overline{b(w,v)}.\]
\begin{remark}
It is not uncommon to see hermitian or Hermitean instead of Hermitian. The name is a tribute to Charles Hermite of the Ecole Polytechnique.
\end{remark}