5.8 $ \mathbb{R}[x]_2$ ¤Ë¤ª¤±¤ëÀþ·ÁÊÑ´¹¤Î¸ÇÍ­¶õ´Ö

Îã 5.21 (¿¹à¼°¤Î¶õ´Ö¤Ë¤ª¤±¤ë¸ÇÍ­ÃÍÌäÂê¤Î¶ñÂÎÎã)   Àþ·ÁÊÑ´¹ $ F:\mathbb{R}[x]_2\to\mathbb{R}[x]_2$;

$\displaystyle g(x)=F(f(x))=f(1+2x)$    

¤Î¸ÇÍ­ÃÍ¡¤¸ÇÍ­¶õ´Ö¤òµá¤á¤ë¡¥

¤Þ¤º¡¤Àþ·ÁÊÑ´¹ $ F$ ¤Îɽ¸½¹ÔÎó $ A$ ¤òµá¤á¤ë¡¥ ´ðÄì $ \Sigma=\{1,\,\,x,\,\,x^2\}$ ¤Î¤â¤È¤Ç ¿¹à¼° $ f(x)$, $ g(x)$ ¤ÎºÂɸ¤ò $ (a_0,a_1,a_2)_{\Sigma}$, $ (b_0,b_1,b_2)_{\Sigma}$ ¤È¤·¤Æɽ¤¹¤È¡¤

$\displaystyle ($¢¤$\displaystyle ) \qquad$ $\displaystyle f(x)=a_0+a_1x+a_2x^2= \left(1,\,\, x,\,\, x^2\right)\begin{bmatrix}a_0 \\ a_1 \\ a_2 \end{bmatrix} = \left(1,\,\, x,\,\, x^2\right)\vec{a},$    
$\displaystyle ($¢¥$\displaystyle ) \qquad$ $\displaystyle g(x)=b_0+b_1x+b_2x^2= \left(1,\,\, x,\,\, x^2\right)\begin{bmatrix}b_0 \\ b_1 \\ b_2 \end{bmatrix} = \left(1,\,\, x,\,\, x^2\right)\vec{b}$    

¤È½ñ¤±¤ë¡¥ ¤³¤Î¤È¤­¡¤$ f$ ¤ò $ F$ ¤Ç¼Ì¤¹¤È

$\displaystyle g(x)$ $\displaystyle =F(f(x))=F(a_0+a_1x+a_2x^2)= a_0F(1)+a_1F(x)+a_2F(x^2)$    
  $\displaystyle = \left(F(1),\,\, F(x),\,\, F(x^2)\right)\begin{bmatrix}a_0 \\ a_1 \\ a_2 \end{bmatrix}= \left(F(1),\,\, F(x),\,\, F(x^2)\right)\vec{a}$    

¤È¤Ê¤ë¡¥ ¤³¤³¤Ç

$\displaystyle \left(F(1),\,\, F(x),\,\, F(x^2)\right)$ $\displaystyle = \left(1,\,\, 1+2x,\,\, (1+2x)^2\right)= \left(1,\,\, 1+2x,\,\, 1+4x+4x^2\right)$    
  $\displaystyle = \left(1,\,\, x,\,\, x^2\right)\begin{bmatrix}1 & 1 & 1 \\ 0 & 2 & 4 \\ 0 & 0 & 4 \\ \end{bmatrix} = \left(1,\,\, x,\,\, x^2\right)A$    

¤Ç¤¢¤ë¤«¤é¡¤ÂåÆþ¤¹¤ë¤È

$\displaystyle ($¡ù$\displaystyle )\qquad g(x)= \left(1,\,\, x,\,\, x^2\right)A \vec{a}$    

¤¬ÆÀ¤é¤ì¤ë¡¥ $ A$ ¤¬ $ F$ ¤Î´ðÄì $ \Sigma$ ¤Ë´Ø¤¹¤ëɽ¸½¹ÔÎó¤Ç¤¢¤ë¡¥ ¤Þ¤¿¡¤¤³¤Î·ë²Ì¤Ï

$\displaystyle g(x)$ $\displaystyle =F(f(x))=f(1+2x)=a_0+a_1(1+2x)+a_2(1+2x)^2$    
  $\displaystyle =(a_0+a_1+a_2)+(2a_1+4a_2)x+4a_2x^2$    
  $\displaystyle = \left(1,\,\, x,\,\, x^2\right)\begin{bmatrix}a_0+a_1+a_2 \\ 2a_...
...matrix}a_0 \\ a_1 \\ a_2 \end{bmatrix} = \left(1,\,\, x,\,\, x^2\right)A\vec{a}$    

¤È¤·¤Æ¤âÆÀ¤é¤ì¤ë¡¥ (¢¥)¤È(¡ù)¤è¤ê¡¤ Àþ·ÁÊÑ´¹ $ F$ ¤Ï

$\displaystyle \varphi:\mathbb{R}^3\to\mathbb{R}^3; \qquad \vec{b}=\varphi(\vec{a})=A\vec{a}$    

¤Ë¤è¤êÄêµÁ¤µ¤ì¤ëÀþ·ÁÊÑ´¹ $ \varphi$ ¤ÈÅù²Á¤Ç¤¢¤ë¡¥

¼¡¤Ë $ F$ ¤Î¸ÇÍ­Ãͤòµá¤á¤ë¡¥ $ F$ ¤Ë´Ø¤¹¤ë¸ÇÍ­ÊýÄø¼°¤Ï $ F(f)=\lambda\,f$ ¤Ç¤¢¤ë¡¥ (¡ù)¤È(¢¤)¤òÍѤ¤¤ë¤È

  $\displaystyle g(x)=F(f(x))= \left(1,\,\, x,\,\, x^2\right)A\vec{a}, \qquad \lambda f(x)= \left(1,\,\, x,\,\, x^2\right)(\lambda\vec{a})$    

¤Èɽ¤µ¤ì¤ë¤Î¤Ç¡¤ ¸ÇÍ­ÊýÄø¼°¤Ï

$\displaystyle (¡ý)\qquad A\vec{a}=\lambda\vec{a}$    

¤ÈÅù²Á¤Ç¤¢¤ë¡¥ ¤Ä¤Þ¤ê¹ÔÎó $ A$ ¤Ë´Ø¤¹¤ë¸ÇÍ­ÃÍÌäÂê¤ò²ò¤±¤Ð¤è¤¤¡¥ $ F$ ¤Î¸Çͭ¿¹à¼° $ g_F(t)$ ¤È $ A$ ¤Î¸Çͭ¿¹à¼° $ g_A(t)$ ¤ÏÅù¤·¤¯¡¤

$\displaystyle g_F(t)=g_A(t)=\det(tE-A)= \begin{vmatrix}t-1 & -1 & -1 \\ 0 & t-2 & -4 \\ 0 & 0 & t-4 \end{vmatrix} = (t-1)(t-2)(t-4)$    

¤È¤Ê¤ë¡¥ $ g_F(\lambda)=0$ ¤è¤ê¸ÇÍ­ÃÍ¤Ï $ \lambda=1,2,4$ ¤Ç¤¢¤ë¡¥

$ F$ ¤Î¸ÇÍ­¶õ´Ö¤òµá¤á¤ë¡¥ ¸ÇÍ­ÊýÄø¼°(¡ý)¤è¤ê $ \varphi$ ¤Î¸ÇÍ­¥Ù¥¯¥È¥ë $ \vec{a}$ ¤òÄê¤á¡¤ ¤½¤Î¸å(¢¤)¤ËÂåÆþ¤· $ F$ ¤Î¸ÇÍ­¥Ù¥¯¥È¥ë $ f(x)$ ¤òÄê¤á¤ì¤Ð¤è¤¤¡¥ $ \lambda=4$ ¤Î¤È¤­¡¤

$\displaystyle 4E-A= \begin{bmatrix}3 & -1 & -1 \\ 0 & 2 & -4 \\ 0 & 0 & 0 \end{...
...ÊÌó²½}}\quad \begin{bmatrix}1 & 0 & -1 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \end{bmatrix}$    

¤È¤Ê¤ë¤Î¤Ç¡¤$ a_1-a_3=0$, $ a_2-2a_3=0$ ¤è¤ê¡¤ $ A$ ¤Î $ \lambda=4$ ¤Ë´Ø¤¹¤ë¸ÇÍ­¥Ù¥¯¥È¥ë¤Ï

$\displaystyle \vec{a}= \begin{bmatrix}a_0 \\ a_1 \\ a_2 \end{bmatrix} = \begin{...
...\\ c \end{bmatrix} = c \begin{bmatrix}1 \\ 2 \\ 1 \end{bmatrix} \qquad (c\neq0)$    

¤ÈÆÀ¤é¤ì¤ë¡¥ ¤è¤Ã¤Æ $ F$ ¤Î $ \lambda=4$ ¤Ë´Ø¤¹¤ë¸ÇÍ­¶õ´Ö¤Ï

$\displaystyle W(4;F)= \left\{\left.\,{c(1+2x+x^2)}\,\,\right\vert\,\,{c\in\math...
...\,\right\} = \left\langle 1+2x+x^2\right\rangle = \left\langle f_1\right\rangle$    

¤È¤Ê¤ë¡¥ $ \lambda=2$ ¤Î¤È¤­¡¤

$\displaystyle 2E-A= \begin{bmatrix}1 & -1 & -1 \\ 0 & 0 & -4 \\ 0 & 0 & -2 \end...
...´ÊÌó²½}}\quad \begin{bmatrix}1 & -1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$    

¤È¤Ê¤ë¤Î¤Ç¡¤$ a_1-a_2=0$, $ a_3=0$ ¤è¤ê¡¤ $ A$ ¤Î $ \lambda=2$ ¤Ë´Ø¤¹¤ë¸ÇÍ­¥Ù¥¯¥È¥ë¤Ï

$\displaystyle \vec{a}= \begin{bmatrix}a_0 \\ a_1 \\ a_2 \end{bmatrix} = \begin{...
...\\ 0 \end{bmatrix} = c \begin{bmatrix}1 \\ 1 \\ 0 \end{bmatrix} \qquad (c\neq0)$    

¤ÈÆÀ¤é¤ì¤ë¡¥ ¤è¤Ã¤Æ $ F$ ¤Î $ \lambda=2$ ¤Ë´Ø¤¹¤ë¸ÇÍ­¶õ´Ö¤Ï

$\displaystyle W(2;F)= \left\{\left.\,{c(1+x)}\,\,\right\vert\,\,{c\in\mathbb{R}}\,\right\} = \left\langle 1+x\right\rangle = \left\langle f_2\right\rangle$    

¤È¤Ê¤ë¡¥ $ \lambda=1$ ¤Î¤È¤­¡¤

$\displaystyle E-A= \begin{bmatrix}0 & -1 & -1 \\ 0 & -1 & -4 \\ 0 & 0 & -3 \end...
...{´ÊÌó²½}}\quad \begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$    

¤È¤Ê¤ë¤Î¤Ç¡¤$ a_2=0$, $ a_3=0$ ¤è¤ê¡¤ $ A$ ¤Î $ \lambda=1$ ¤Ë´Ø¤¹¤ë¸ÇÍ­¥Ù¥¯¥È¥ë¤Ï

$\displaystyle \vec{a}= \begin{bmatrix}a_0 \\ a_1 \\ a_2 \end{bmatrix} = \begin{...
...\\ 0 \end{bmatrix} = c \begin{bmatrix}1 \\ 0 \\ 0 \end{bmatrix} \qquad (c\neq0)$    

¤ÈÆÀ¤é¤ì¤ë¡¥ ¤è¤Ã¤Æ $ F$ ¤Î $ \lambda=1$ ¤Ë´Ø¤¹¤ë¸ÇÍ­¶õ´Ö¤Ï

$\displaystyle W(1;F)= \left\{\left.\,{c}\,\,\right\vert\,\,{c\in\mathbb{R}}\,\right\} = \left\langle 1\right\rangle = \left\langle f_3\right\rangle$    

¤È¤Ê¤ë¡¥

¸ÇÍ­¶õ´Ö $ W(4)$, $ W(2)$, $ W(1)$ ¤Î¤½¤ì¤¾¤ì¤Î´ðÄì¤Ï $ \{f_1\}$, $ \{f_2\}$, $ \{f_3\}$ ¤Ç¤¢¤ë¡¥ ¤³¤ì¤é¤Ï

$\displaystyle ($¡ö$\displaystyle )\qquad \left(f_1,\,\,f_2,\,\,f_3\right)= \left(1+2x+x^2,\,\, 1+x...
...1 & 1 \\ 2 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix} = \left(1,\,\, x,\,\, x^2\right)P$    

¤Èɽ¤µ¤ì¤ë¡¥ $ \det(P)=1\neq0$ ¤è¤ê $ \{f_1,f_2,f_3\}$ ¤Ï 1 ¼¡ÆÈΩ¤È¤Ê¤ë¡¥ ¤è¤Ã¤Æ¡¤

  $\displaystyle W(4)\cap W(2)= \left\langle f_1 \right\rangle \cap \left\langle f...
...left\langle f_1 \right\rangle \cap \left\langle f_3 \right\rangle =\{\vec{0}\},$    
  $\displaystyle W(2)\cap W(1)= \left\langle f_2 \right\rangle \cap \left\langle f_3 \right\rangle =\{\vec{0}\}$    

¤È¤Ê¤ê¡¤

$\displaystyle W(4)\oplus W(2)\oplus W(1)= \left\langle f_1 \right\rangle \oplus...
...right\rangle = \left\langle f_1,\,\, f_2,\,\, f_3\right\rangle =\mathbb{R}[x]_2$    

¤¬À®¤êΩ¤Ä¡¥ $ \mathbb{R}[x]_2$ ¤Ï $ W(4)$, $ W(2)$, $ W(1)$ ¤ËľÏÂʬ²ò¤µ¤ì¤ë¡¥ $ \{f_1,f_2,f_3\}$ ¤Ï $ \mathbb{R}[x]_2$ ¤Î´ðÄì¤È¤Ê¤ë¡¥

Àþ·ÁÊÑ´¹ $ F$ ¤Î´ðÄì $ \Sigma=\{1,\,\,x,\,\,x^2\}$ ¤Ë¤ª¤±¤ë ɽ¸½¹ÔÎó¤Ï $ A$ ¤Ç¤¢¤ë¡¥ ´ðÄì $ \Sigma'=\{f_1,\,\,f_2,\,\,f_3\}$ ¤Ë¤ª¤±¤ë $ F$ ¤Îɽ¸½¹ÔÎó¤òµá¤á¤ë¡¥ ´ðÄì $ \Sigma'$ ¤Ë¤ª¤±¤ë $ f$ ¤ÎºÂɸ¤ò $ (\tilde{a}_0,\tilde{a}_1,\tilde{a}_2)_{\Sigma'}$ ¤È¤¹¤ë¤È¡¤ ´ðÄìÊÑ´¹(¡ö)¤òÍѤ¤¤Æ¡¤

  $\displaystyle f(x)= a_0+a_1x+a_2x^2= \tilde{a}_1\,f_1(x)+ \tilde{a}_2\,f_2(x)+ \tilde{a}_3\,f_3(x)$    
  $\displaystyle \quad\Rightarrow\quad \left(1,\,\,x,\,\,x^2\right)\begin{bmatrix}...
...1,\,\,x,\,\,x^2\right)\vec{a} = \left(f_1,\,\,f_2,\,\,f_3\right)\tilde{\vec{a}}$    
  $\displaystyle \quad\Rightarrow\quad \left(1,\,\,x,\,\,x^2\right)\vec{a} = \left...
...x,\,\,x^2\right)P\tilde{\vec{a}} \quad\Rightarrow\quad \vec{a}=P\tilde{\vec{a}}$    

¤Èɽ¤µ¤ì¤ë¡¥ $ \vec{a}=P\tilde{\vec{a}}$ ¤Ï $ \Sigma$ ¤«¤é $ \Sigma'$ ¤Ø¤ÎºÂɸÊÑ´¹¤Ç¤¢¤ë¡¥ $ g$ ¤Î $ \Sigma'$ ¤Ë¤ª¤±¤ëºÂɸ¤ò $ (\tilde{b}_0,\tilde{b}_1,\tilde{b}_2)_{\Sigma'}$ ¤È¤¹¤ë¤È¡¤ ƱÍͤˤ·¤Æ $ \vec{b}=P\tilde{\vec{b}}$ ¤¬À®¤êΩ¤Ä¡¥ Àþ·ÁÊÑ´¹ $ F$ ¤ÏÀþ·ÁÊÑ´¹ $ \varphi:\,\vec{b}=A\vec{a}$ ¤ÈÅù²Á¤Ç¤¢¤ë¤«¤é¡¤ $ \vec{b}=A\vec{a}$ ¤ËºÂɸÊÑ´¹ $ \vec{a}=P\tilde{\vec{a}}$, $ \vec{b}=P\tilde{\vec{b}}$ ¤òÂåÆþ¤·¤Æ

$\displaystyle \vec{b}=A\vec{a} \quad\Rightarrow\quad P\tilde{\vec{b}}=AP\tilde{...
...ec{a}} \quad\Rightarrow\quad \tilde{\vec{b}}=D\tilde{\vec{a}}, \quad D=P^{-1}AP$    

¤òÆÀ¤ë¡¥ ´ðÄì $ \Sigma'=\{f_1,\,\,f_2,\,\,f_3\}=\{1+2x+x^2,\,\,1+x,\,\,1\}$ ¤Ë´Ø¤¹¤ë $ F$ ¤Îɽ¸½¹ÔÎó¤Ï

$\displaystyle D=P^{-1}AP= \begin{bmatrix}1 & 1 & 1 \\ 2 & 1 & 0 \\ 1 & 0 & 0 \e...
...end{bmatrix} = \begin{bmatrix}4 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{bmatrix}$    

¤È¤Ê¤ë¡¥

Ãí°Õ 5.22 (°ìÈ̤Υ٥¯¥È¥ë¶õ´Ö¤Î¸ÇÍ­ÃÍÌäÂê)   °ìÈ̤Υ٥¯¥È¥ë¶õ´Ö¤Ë¤ª¤±¤ë¸ÇÍ­ÃÍÌäÂê¤Ï ¿ô¥Ù¥¯¥È¥ë¶õ´Ö( $ \mathbb{R}^n$ ¤Þ¤¿¤Ï $ \mathbb{C}^n$)¤Ë¤ª¤±¤ë ¸ÇÍ­ÃÍÌäÂê¤Ë´Ô¸µ¤·¤ÆµÄÏÀ¤¹¤ì¤Ð¤è¤¤¡¥ ¾å¤ÎÎãÂê¤Ç¤Ï¼¡¤Î¤è¤¦¤ËƱ°ì»ë¤ò¹Ô¤Ã¤¿¡§

Àþ·ÁÊÑ´¹ $ F:\mathbb{R}[x]_2\to\mathbb{R}[x]_2$$\displaystyle \quad$ $\displaystyle \Leftrightarrow$   Àþ·ÁÊÑ´¹ $ \varphi:\mathbb{R}^3\to\mathbb{R}^3$    
Àþ·ÁÊÑ´¹ $ g=F(f)$$\displaystyle \quad$ $\displaystyle \Leftrightarrow$   Àþ·ÁÊÑ´¹ $ \vec{b}=\varphi(\vec{a})$$\displaystyle \quad \Leftrightarrow$   Àþ·ÁÊÑ´¹ $ \vec{b}=A\vec{a}$    
$\displaystyle f(x) \quad$ $\displaystyle \Leftrightarrow\quad \vec{a}$    
$\displaystyle g(x) \quad$ $\displaystyle \Leftrightarrow\quad \vec{b}$    
$ F$ ¤Îɽ¸½¹ÔÎó $ A$$\displaystyle \quad$ $\displaystyle =$   $ \varphi$ ¤Îɽ¸½¹ÔÎó $ A$    
¸ÇÍ­ÊýÄø¼° $ F(f)=\lambda\,f$$\displaystyle \quad$ $\displaystyle \Leftrightarrow$   ¸ÇÍ­ÊýÄø¼° $ \varphi(\vec{a})=\lambda\vec{a}$$\displaystyle \quad \Leftrightarrow$   ¸ÇÍ­ÊýÄø¼° $ A\vec{a}=\lambda\vec{a}$    
$ F$ ¤Î¸Çͭ¿¹à¼° $ g_F(t)$$\displaystyle \quad$ $\displaystyle =$   $ \varphi$ ¤Î¸Çͭ¿¹à¼° $ g_\varphi(t)$$\displaystyle \quad=$   $ A$ ¤Î¸Çͭ¿¹à¼° $ g_A(t)$    
$ F$ ¤Î¸ÇÍ­ÃÍ $ \lambda$$\displaystyle \quad$ $\displaystyle =$   $ \varphi$ ¤Î¸ÇÍ­ÃÍ $ \lambda$$\displaystyle \quad=$   $ A$ ¤Î¸ÇÍ­ÃÍ $ \lambda$    
¸ÇÍ­¶õ´Ö $ W(\lambda;F)$$\displaystyle \quad$ $\displaystyle \Leftrightarrow$   ¸ÇÍ­¶õ´Ö $ W(\lambda;\varphi)$    

Ãí°Õ 5.23 (°ìÈ̤Υ٥¯¥È¥ë¶õ´Ö¤Ë¤ª¤±¤ëÀþ·ÁÊÑ´¹)   ¤É¤Î¤è¤¦¤Ê¥Ù¥¯¥È¥ë¶õ´Ö¤Ë¤ª¤±¤ë Àþ·ÁÊÑ´¹¤Ç¤¢¤Ã¤Æ¤â¹ÔÎó $ A$ ¤ÈƱ°ì»ë¤¹¤ì¤Ð¤è¤¤¡¥


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