이와사와 분해 (Iwasawa decomposition)

개요

• $G=KAN$ 또는 $G=NAK$

예

• $G=\operatorname{SL}(2,R)$

$$ K=\left\{\left( \begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{array} \right)|\theta\in \mathbb{R}\right\} $$ $$A=\left\{\left( \begin{array}{cc} a & 0 \\ 0 & a^{-1} \\ \end{array} \right) | a\in \mathbb{R}^{+}\right\} $$ $$ N=\left\{\left( \begin{array}{cc} 1 & x \\ 0 & 1 \\ \end{array} \right)|x\in \mathbb{R}\right\} $$

• 푸앵카레 상반평면 모델에서 점 $x+i y\in \mathbb{H}=G/K$는 다음에 대응된다

$$ \left( \begin{array}{cc} 1 & x \\ 0 & 1 \\ \end{array} \right)\left( \begin{array}{cc} \sqrt{y} & 0 \\ 0 & \frac{1}{\sqrt{y}} \\ \end{array} \right)K $$

사전형태의 자료

• http://en.wikipedia.org/wiki/Iwasawa_decomposition