実対称行列を回転行列(実直交行列)と実対角行列で表現する

$$\begin{array}{rcl}{S}_2& =& \left[\begin{array}{cc}x& y\\ y& z\end{array}\right]\cdots S_2:x,y,z\mathrm{を}\mathrm{実}\mathrm{数}\mathrm{と}\mathrm{す}\mathrm{る}2\times 2\mathrm{の}\mathrm{実}\mathrm{対}\mathrm{称}\mathrm{行}\mathrm{列}\\ & & \cdots \mathrm{実}\mathrm{対}\mathrm{称}\mathrm{行}\mathrm{列}:S_2=S_2^T(\mathrm{対}\mathrm{称}\mathrm{行}\mathrm{列}),S_2=S_2^{\ast }(\mathrm{実}\mathrm{行}\mathrm{列}(\mathrm{各}\mathrm{要}\mathrm{素}\mathrm{の}\mathrm{複}\mathrm{素}\mathrm{共}\mathrm{役}\mathrm{が}\mathrm{変}\mathrm{わ}\mathrm{ら}\mathrm{な}\mathrm{い},\mathrm{虚}\mathrm{部}\mathrm{が}0))\\ \\ R(\theta )& =& \left[\begin{array}{cc}\cos \left(\theta \right)& -\sin \left(\theta \right)\\ \sin \left(\theta \right)& \cos \left(\theta \right)\end{array}\right]\\ R(\theta {)}^T& =& \left[\begin{array}{cc}\cos \left(\theta \right)& \sin \left(\theta \right)\\ -\sin \left(\theta \right)& \cos \left(\theta \right)\end{array}\right]=\left[\begin{array}{cc}\cos (-\theta )& -\sin (-\theta )\\ \sin (-\theta )& \cos (-\theta )\end{array}\right]=R(-\theta )\cdots \mathrm{逆}\mathrm{回}\mathrm{転}\\ R(\theta )R(\theta {)}^T& =& R(\theta {)}^{T}R(\theta )=I\cdots R(\theta ):\mathrm{直}\mathrm{交}\mathrm{行}\mathrm{列}(\mathrm{特}\mathrm{に}\mathrm{各}\mathrm{要}\mathrm{素}\mathrm{の}\mathrm{複}\mathrm{素}\mathrm{共}\mathrm{役}\mathrm{が}\mathrm{変}\mathrm{わ}\mathrm{ら}\mathrm{な}\mathrm{い}\mathrm{の}\mathrm{で}\mathrm{実}\mathrm{直}\mathrm{交}\mathrm{行}\mathrm{列})\\ \\ \mathrm{\Lambda }& =& \left[\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right]\cdots {\lambda }_1,{\lambda }_2\mathrm{は}{S}_2\mathrm{の}\mathrm{固}\mathrm{有}\mathrm{値}\\ & & \cdots \mathrm{対}\mathrm{角}\mathrm{行}\mathrm{列}(\mathrm{特}\mathrm{に}\mathrm{各}\mathrm{要}\mathrm{素}\mathrm{の}\mathrm{複}\mathrm{素}\mathrm{共}\mathrm{役}\mathrm{が}\mathrm{変}\mathrm{わ}\mathrm{ら}\mathrm{な}\mathrm{い}\mathrm{場}\mathrm{合}，\mathrm{実}\mathrm{対}\mathrm{角}\mathrm{行}\mathrm{列})\end{array}$$ $$\begin{array}{rcl}{S}_2& =& R(\theta )\mathrm{\Lambda }R(\theta {)}^T\\ & =& \left[\begin{array}{cc}\cos \left(\theta \right)& -\sin \left(\theta \right)\\ \sin \left(\theta \right)& \cos \left(\theta \right)\end{array}\right]\left[\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right]\left[\begin{array}{cc}\cos \left(\theta \right)& \sin \left(\theta \right)\\ -\sin \left(\theta \right)& \cos \left(\theta \right)\end{array}\right]\\ & =& \left[\begin{array}{cc}{\lambda }_1\cos \left(\theta \right)& -{\lambda }_2\sin \left(\theta \right)\\ {\lambda }_1\sin \left(\theta \right)& {\lambda }_2\cos \left(\theta \right)\end{array}\right]\left[\begin{array}{cc}\cos \left(\theta \right)& \sin \left(\theta \right)\\ -\sin \left(\theta \right)& \cos \left(\theta \right)\end{array}\right]\\ & =& \left[\begin{array}{cc}{\lambda }_1{\cos }^2\left(\theta \right)+{\lambda }_2{\sin }^2\left(\theta \right)& ({\lambda }_1-{\lambda }_2)\cos \left(\theta \right)\sin \left(\theta \right)\\ ({\lambda }_1-{\lambda }_2)\cos \left(\theta \right)\sin \left(\theta \right)& {\lambda }_1{\sin }^2\left(\theta \right)+{\lambda }_2{\cos }^2\left(\theta \right)\end{array}\right]\\ \\ x& =& {\lambda }_1{\cos }^2\left(\theta \right)+{\lambda }_2{\sin }^2\left(\theta \right)\\ & =& \frac{1}{2}2\{{\lambda }_1{\cos }^2\left(\theta \right)+{\lambda }_2{\sin }^2\left(\theta \right)\}\\ & =& \frac{1}{2}\{2{\lambda }_1{\cos }^2\left(\theta \right)+2{\lambda }_2{\sin }^2\left(\theta \right)\}\\ & =& \frac{1}{2}\{{\lambda }_1({\cos }^2\left(\theta \right)+1-{\sin }^2\left(\theta \right))+{\lambda }_2({\sin }^2\left(\theta \right)+1-{\cos }^2\left(\theta \right))\}\cdots {\cos }^2\left(\theta \right)=1-{\sin }^2\left(\theta \right),{\sin }^2\left(\theta \right)=1-{\cos }^2\left(\theta \right)\\ & =& \frac{1}{2}\{{\lambda }_1({\cos }^2\left(\theta \right)+1-{\sin }^2\left(\theta \right))+{\lambda }_2(1-({\cos }^2\left(\theta \right)-{\sin }^2\left(\theta \right)))\}\\ & =& \frac{1}{2}\{{\lambda }_1(\cos \left(2\theta \right)+1)+{\lambda }_2(1-\cos \left(2\theta \right))\}\cdots \cos \left(2\theta \right)={\cos }^2\left(\theta \right)-{\sin }^2\left(\theta \right)\\ & =& \frac{1}{2}\{{\lambda }_1\cos \left(2\theta \right)+{\lambda }_1+{\lambda }_2-{\lambda }_2\cos \left(2\theta \right)\}\\ & =& \frac{1}{2}\{({\lambda }_1-{\lambda }_2)\cos \left(2\theta \right)+{\lambda }_1+{\lambda }_2\}\\ y& =& ({\lambda }_1-{\lambda }_2)\cos \left(\theta \right)\sin \left(\theta \right)\\ & =& \frac{1}{2}2({\lambda }_1-{\lambda }_2)\cos \left(\theta \right)\sin \left(\theta \right)\\ & =& \frac{1}{2}({\lambda }_1-{\lambda }_2)\sin \left(2\theta \right)\cdots \sin \left(2\theta \right)=2\cos \left(\theta \right)\sin \left(\theta \right)\\ z& =& {\lambda }_1{\sin }^2\left(\theta \right)+{\lambda }_2{\cos }^2\left(\theta \right)\\ & =& \frac{1}{2}2\{{\lambda }_1{\sin }^2\left(\theta \right)+{\lambda }_2{\cos }^2\left(\theta \right)\}\\ & =& \frac{1}{2}\{2{\lambda }_1{\sin }^2\left(\theta \right)+2{\lambda }_2{\cos }^2\left(\theta \right)\}\\ & =& \frac{1}{2}\{{\lambda }_1({\sin }^2\left(\theta \right)+1-{\cos }^2\left(\theta \right))+{\lambda }_2({\cos }^2\left(\theta \right)+1-{\sin }^2\left(\theta \right))\}\cdots {\sin }^2\left(\theta \right)=1-{\cos }^2\left(\theta \right),{\cos }^2\left(\theta \right)=1-{\sin }^2\left(\theta \right)\\ & =& \frac{1}{2}\{{\lambda }_1(1-({\cos }^2\left(\theta \right)-{\sin }^2\left(\theta \right)))+{\lambda }_2({\cos }^2\left(\theta \right)+1-{\sin }^2\left(\theta \right))\}\\ & =& \frac{1}{2}\{{\lambda }_1(1-\cos \left(2\theta \right))+{\lambda }_2(\cos \left(2\theta \right)+1)\}\cdots \cos \left(2\theta \right)={\cos }^2\left(\theta \right)-{\sin }^2\left(\theta \right)\\ & =& \frac{1}{2}\{{\lambda }_1-{\lambda }_1\cos \left(2\theta \right)+{\lambda }_2\cos \left(2\theta \right)+{\lambda }_2\}\\ & =& \frac{1}{2}\{-({\lambda }_1-{\lambda }_2)\cos \left(2\theta \right)+{\lambda }_1+{\lambda }_2\}\end{array}$$ $$\{\begin{array}{rcl}x& =& \frac{1}{2}\{({\lambda }_1-{\lambda }_2)\cos \left(2\theta \right)+{\lambda }_1+{\lambda }_2\}\\ y& =& \frac{1}{2}({\lambda }_1-{\lambda }_2)\sin \left(2\theta \right)\\ z& =& \frac{1}{2}\{-({\lambda }_1-{\lambda }_2)\cos \left(2\theta \right)+{\lambda }_1+{\lambda }_2\}\end{array}$$ $$\begin{array}{rcl}2x& =& ({\lambda }_1-{\lambda }_2)\cos \left(2\theta \right)+{\lambda }_1+{\lambda }_2\\ 2z& =& -({\lambda }_1-{\lambda }_2)\cos \left(2\theta \right)+{\lambda }_1+{\lambda }_2\\ 2x-2z& =& ({\lambda }_1-{\lambda }_2)\cos \left(2\theta \right)+{\lambda }_1+{\lambda }_2-\{-({\lambda }_1-{\lambda }_2)\cos \left(2\theta \right)+{\lambda }_1+{\lambda }_2\}\\ 2(x-z)& =& ({\lambda }_1-{\lambda }_2)\cos \left(2\theta \right)+{\lambda }_1+{\lambda }_2+({\lambda }_1-{\lambda }_2)\cos \left(2\theta \right)-{\lambda }_1-{\lambda }_2\\ 2(x-z)& =& 2({\lambda }_1-{\lambda }_2)\cos \left(2\theta \right)\\ \cos \left(2\theta \right)& =& \frac{x-z}{{\lambda }_1-{\lambda }_2}=\frac{x-z}{\sqrt{(x-z{)}^2+4y^2}}\cdots {\lambda }_1-{\lambda }_2=\sqrt{(x-z{)}^2+4y^2}\\ \sin \left(2\theta \right)& =& \frac{2y}{{\lambda }_1-{\lambda }_2}=\frac{2y}{\sqrt{(x-z{)}^2+4y^2}}\cdots {\lambda }_1-{\lambda }_2=\sqrt{(x-z{)}^2+4y^2}\end{array}$$ ${\lambda }_1,{\lambda }_2$だけでなく$\theta$も$x,y,z$により一意に決まる．