5.25 3 ¼¡ÂоιÔÎó¤ÎÂгѲ½

Îã 5.74 (ÂоιÔÎó¤ÎÂгѲ½¤Î¶ñÂÎÎã)   ÂоιÔÎó

$\displaystyle A= \begin{bmatrix}1 & 2 & -1 \\ 2 & -2 & 2 \\ -1 & 2 & 1 \end{bmatrix}$    

¤òľ¸ò¹ÔÎó¤ÇÂгѲ½¤¹¤ë¡¥ $ A$ ¤Î¸Çͭ¿¹à¼°¤Ï

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

¤Ç¤¢¤ë¤«¤é¡¤ ¸ÇÍ­ÃÍ¤Ï $ g_A(\lambda)=0$ ¤è¤ê $ \lambda=2$(2 ½Å),$ -4$ ¤È¤Ê¤ë¡¥

  $\displaystyle 2E-A= \begin{bmatrix}1 & -2 & 1 \\ -2 & 4 & -2 \\ 1 & -2 & 1 \end...
...ÊÌó²½}}\quad \begin{bmatrix}1 & -2 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix},$    
  $\displaystyle -4E-A= \begin{bmatrix}-5 & -2 & 1 \\ -2 & -2 & -2 \\ 1 & -2 & -5 ...
...´ÊÌó²½}}\quad \begin{bmatrix}1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{bmatrix}$    

¤è¤ê¡¤$ \lambda=2$ ¤È $ \lambda=-4$ ¤Ë °¤¹¤ë¸ÇÍ­¥Ù¥¯¥È¥ë¤Ï¤½¤ì¤¾¤ì

  $\displaystyle \vec{x}= \begin{bmatrix}2x_2-x_3 \\ x_2 \\ x_3 \end{bmatrix} = \b...
...x} + c_2 \begin{bmatrix}-1 \\ 0 \\ 1 \end{bmatrix} = c_1\vec{p}_1+c_2\vec{p}_2,$    
  $\displaystyle \vec{x}= \begin{bmatrix}x_3 \\ -2x_3 \\ x_3 \end{bmatrix} = \begi...
... \\ c \end{bmatrix} = c \begin{bmatrix}1 \\ -2 \\ 1 \end{bmatrix} = c \vec{p}_3$    

¤È¤Ê¤ë¡¥ $ \{\vec{p}_1,\vec{p}_2,\vec{p}_3\}$ ¤Ï1 ¼¡ÆÈΩ¤Ç¤¢¤ë¤«¤é¡¤ $ P=\begin{bmatrix}\vec{p}_1 & \vec{p}_2 & \vec{p}_3 \end{bmatrix}$ ¤ÏÀµÂ§¤È¤Ê¤ë¡¥ ¤·¤«¤· $ P$ ¤Ïľ¸ò¹ÔÎó¤Ç¤Ï¤Ê¤¤¤Î¤Ç¡¤ ľ¸ò¹ÔÎó¤È¤Ê¤ë¤è¤¦¤Ë¸ÇÍ­¥Ù¥¯¥È¥ë¤òÁª¤Óľ¤¹¡¥

$\displaystyle \left({\vec{p}_1}\,,\,{\vec{p}_2}\right)=-2\neq0, \quad \left({\v...
...}_1}\,,\,{\vec{p}_3}\right)=0, \quad \left({\vec{p}_2}\,,\,{\vec{p}_3}\right)=0$    

¤Ç¤¢¤ë¤«¤é¡¤ $ \vec{p}_1\perp\vec{p}_3$, $ \vec{p}_2\perp\vec{p}_3$ ¤È¤Ê¤ë¡¥ $ \vec{p}_1$, $ \vec{p}_2$ ¤ò¥°¥é¥à¡¦¥·¥å¥ß¥Ã¥È¤Îľ¸ò²½Ë¡¤Ç Àµµ¬Ä¾¸ò²½¤·¡¤$ \vec{p}_3$ ¤ÏÀµµ¬²½¤¹¤ì¤Ð¤è¤¤¡¥

$\displaystyle \vec{q}_1$ $\displaystyle = \frac{\vec{p}_1}{\Vert\vec{p}_1\Vert}= \frac{1}{\sqrt{5}} \begin{bmatrix}2 \\ 1 \\ 0 \end{bmatrix},$    
$\displaystyle \vec{p}'_2$ $\displaystyle = \vec{p}_2-\left({\vec{p}_1}\,,\,{\vec{q}_1}\right)\vec{q}_1$    
  $\displaystyle = \begin{bmatrix}-1 \\ 0 \\ 1 \end{bmatrix} - \left({\begin{bmatr...
...\ 1 \\ 0 \end{bmatrix} = \frac{1}{5} \begin{bmatrix}-1 \\ 2 \\ 5 \end{bmatrix},$    
$\displaystyle \vec{q}_2$ $\displaystyle = \frac{\vec{p}'_2}{\Vert\vec{p}'_2\Vert}= \frac{1}{\sqrt{30}} \begin{bmatrix}-1 \\ 2 \\ 5 \end{bmatrix},$    
$\displaystyle \vec{q}_3$ $\displaystyle = \frac{\vec{p}_3}{\Vert\vec{p}_3\Vert}= \frac{1}{\sqrt{6}} \begin{bmatrix}1 \\ -2 \\ 1 \end{bmatrix}$    

¤È¤ª¤¯¤È¡¤ $ \left({\vec{q}_i}\,,\,{\vec{q}_j}\right)=\delta_{ij}$($ i,j=1,2,3$) ¤¬À®¤êΩ¤ÁÀµµ¬Ä¾¸ò·Ï¤È¤Ê¤ë¡¥ °Ê¾å¤è¤ê¹ÔÎó $ A$ ¤Ï

  $\displaystyle D=Q^{-1}AQ={Q}^{T}AQ,$    
  $\displaystyle D=\mathrm{diag}\,(2,2,-4)= \begin{bmatrix}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -4 \end{bmatrix},$    
  $\displaystyle Q= \begin{bmatrix}\vec{p}_1 & \vec{p}_2 & \vec{p}_3 \end{bmatrix}...
...frac{2}{\sqrt{6}} \\ 0 & \frac{5}{\sqrt{30}} & \frac{1}{\sqrt{6}} \end{bmatrix}$    

¤Èľ¸ò¹ÔÎó $ Q$ ¤ÇÂгѲ½¤µ¤ì¤ë¡¥

Ãí°Õ 5.75 (ÂоιÔÎó¤Î¸ÇÍ­¶õ´Ö)   Àþ·ÁÊÑ´¹ $ f:\mathbb{R}^3\to\mathbb{R}^3$; $ \vec{y}=A\vec{x}$ ¤Î ¸ÇÍ­¶õ´Ö¤Ï

$\displaystyle W(2;f)= \left\langle \vec{p}_1,\,\, \vec{p}_2\right\rangle = \lef...
...4;f)= \left\langle \vec{p}_3\right\rangle = \left\langle \vec{q}_3\right\rangle$    

¤Ç¤¢¤ë¡¥ $ \dim(W(2;f))=2$, $ \dim(W(-4;f))=1$, $ W(2;f)\cap W(-4;f)=\{\vec{0}\}$ ¤è¤ê

$\displaystyle ($¡ù$\displaystyle )\qquad W(2;f)\oplus W(-4;f)=\mathbb{R}^3, \qquad \dim(W(2;f))+\dim(W(-4;f))=\dim(\mathbb{R}^3)$    

¤òÆÀ¤ë¡¥ ¸ÇÍ­¶õ´Ö $ W(2;f)$, $ W(-4;f)$ ¤Ï $ \mathbb{R}^3$ ¤ÎľÏÂʬ²ò¤Ç¤¢¤ë¡¥ ¤Þ¤¿¡¤ °Û¤Ê¤ë¸ÇÍ­ÃͤË°¤¹¤ë¸ÇÍ­¥Ù¥¯¥È¥ë¤Ïľ¸ò¤¹¤ë¤Î¤Ç¡¤ ¸ÇÍ­¶õ´Ö¤âľ¸ò¤· $ W(2;f)\perp W(-4;f)$ ¤òÆÀ¤ë¡¥ $ W(2;f)\perp W(-4;f)$ ¤È(¡ù)¤è¤ê¡¤ $ W(2;f)$ ¤Ï $ \mathbb{R}^3$ ¤Ë¤ª¤±¤ë $ W(-4;f)$ ¤Îľ¸òÊä¶õ´Ö¤È¤Ê¤ë¡¥ ¤Þ¤¿µÕ¤Ë $ W(-4;f)$ ¤Ï $ \mathbb{R}^3$ ¤Ë¤ª¤±¤ë $ W(2;f)$ ¤Îľ¸òÊä¶õ´Ö¤È¤Ê¤ë¡¥


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