Hermitian form HermitianForm 2002-02-22 03:11:42 2009-01-31 15:25:54 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \renewcommand{\v}{\mathbf{v}} \newcommand{\w}{\mathbf{w}} A \emph{sesquilinear form} over a pair of complex vector spaces $(V,W)$ is a function $B\colon V \times W \to \mathbb{C}$ satisfying the following properties: \begin{enumerate} \item $B(\v_1+\v_2,\w) = B(\v_1,\w) + B(\v_2,\w)$ \item $B(\v,\w_1+\w_2) = B(\v,\w_1) + B(\v,\w_2)$ \item $B(c\v, d\w) = c B(\v,\w) \overline{d}$ \end{enumerate} for all $\v,\v_1,\v_2 \in V$, $\w, \w_1, \w_2 \in W$, and $c,d \in \mathbb{C}$. The vector spaces $V$ and $W$ are often identical, although the definition does not require them to be the same vector space. A sesquilinear form $B\colon V \times V \to \mathbb{C}$ over a single vector space $V$ is called a \emph{Hermitian form} if it is complex conjugate symmetric: namely, if $B(\v_1,\v_2) = \overline{B(\v_2,\v_1)}$. An inner product over a complex vector space is a positive definite Hermitian form.