OrthogonalGroup 2002-02-22 03:30:32 2006-04-05 01:55:24 Definition orthogonal transformation % 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 Let $Q$ be a non-degenerate symmetric bilinear form over the real vector space $\mathbb{R}^n$. A linear transformation $T\colon V \to V$ is said to \emph{preserve} $Q$ if $Q(Tx,Ty) = Q(x,y)$ for all vectors $x,y \in V$. The subgroup of the general linear group $\operatorname{GL}(V)$ consisting of all linear transformations that preserve $Q$ is called the \emph{orthogonal group} with respect to $Q$, and denoted $\operatorname{O}(n,Q)$. If $Q$ is also positive definite (i.e., $Q$ is an inner product), then $\operatorname{O}(n,Q)$ is equivalent to the group of invertible linear transformations that preserve the standard inner product on $\mathbb{R}^n$, and in this case the group $\operatorname{O}(n,Q)$ is usually denoted $\operatorname{O}(n)$. Elements of $\operatorname{O}(n)$ are called \emph{orthogonal transformations}. One can show that a linear transformation $T$ is an orthogonal transformation if and only if $T^{-1} = T^{\operatorname{T}}$ (i.e., the inverse of $T$ equals the transpose of $T$).