rotation matrixRotationMatrix2005-02-19 10:02:402006-06-13 08:18:38Definition% 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}
\usepackage{amsthm}
\usepackage{mathrsfs}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}}
\usepackage{bbm}
\newcommand{\Z}{\mathbbmss{Z}}
\newcommand{\C}{\mathbbmss{C}}
\newcommand{\R}{\mathbbmss{R}}
\newcommand{\Q}{\mathbbmss{Q}}
\newcommand*{\norm}[1]{\lVert #1 \rVert}
\newcommand*{\abs}[1]{| #1 |}
\newtheorem{thm}{Theorem}
\newtheorem{defn}{Definition}
\newtheorem{prop}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}\begin{defn} A \emph{rotation matrix} is a
(real) orthogonal matrix whose determinant is $+1$.
All $n\times n$ rotation matrices form a group called
the \emph{specia{l} orthogona{l} grou{p}} and it is denoted by
$\operatorname{SO}(n)$.
\end{defn}
\subsubsection*{Examples}
\begin{enumerate}
\item The identity matrix in $\R^n$ is a rotation matrix.
\item The most general rotation matrix in $\R^2$ can be written as
$$
\begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{pmatrix},
$$
where $\theta\in \R$.
Multiplication (from the left) with this matrix
rotates a vector (in $\R^2$) $\theta$ radians in the anti-clockwise
direction.
\end{enumerate}
\subsubsection*{Properties}
\begin{enumerate}
\item Suppose $v\in \R^n$ is a unit vector.
Then there exists a rotation matrix $R$
such that $R\cdot v = (1,0,\ldots, 0)$.
\item
In fact, for $v\in \R^n$, $n\ge 3$, there are many rotation matrices
$\mathbf{R} \in \operatorname{SO}(n)$ such that
$R\cdot v = (1,0,\ldots, 0)^T$.
To see this, let $f$ be the mapping
$f\colon \operatorname{SO}(n-1)\rightarrow \operatorname{SO}(n)$,
defined as
$$
f(Q)=
\begin{pmatrix}
1 & 0_{1\times n-1}\\
0_{n-1\times 1} & Q_{n-1\times n-1}
\end{pmatrix}.
$$
Then for each $Q\in \operatorname{SO}(n-1)$, $f(Q)$
maps $(1,0,\ldots, 0)^T$ onto itself. Thus, if
$R_0 \in \operatorname{SO}(n)$ satisfies $R\cdot v=(1,0,\ldots, 0)^T$,
then $f(Q)\cdot R$ satisfies the same property for all
$Q\in \operatorname{SO}(n-1)$.
\end{enumerate}