5.25 3 慼覆憊墊昫及覆剩祭

拺 5.74 (覆憊墊昫及覆剩祭及填蟲拺) 覆憊墊昫

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

毛躂跟墊昫匹覆剩祭允月‘ 及裔肣聶嫩摯反

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

匹丐月井日﹞ 裔肣襖反 $g_A(\lambda)=0$ 方曰 $\lambda=2$ (2 褐), 午卅月‘

	$\displaystyle 2E-A= \begin{bmatrix}1 & -2 & 1 \\ -2 & 4 & -2 \\ 1 & -2 & 1 \end... ...帎馦罷}\quad \begin{bmatrix}1 & -2 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix},$
	$\displaystyle -4E-A= \begin{bmatrix}-5 & -2 & 1 \\ -2 & -2 & -2 \\ 1 & -2 & -5 ... ...棵沶祭}}\quad \begin{bmatrix}1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{bmatrix}$

方曰﹞ $\lambda=2$ 午 $\lambda=-4$ 卞簞允月裔肣矛弁玄伙反公木冗木

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

午卅月‘ $\{\vec{p}_1,\vec{p}_2,\vec{p}_3\}$ 反1 慼卩峏匹丐月井日﹞ $P=\begin{bmatrix}\vec{p}_1 & \vec{p}_2 & \vec{p}_3 \end{bmatrix}$ 反懇禮午卅月‘ 仄井仄反躂跟墊昫匹反卅中及匹﹞ 躂跟墊昫午卅月方丹卞裔肣矛弁玄伙毛薊太躂允‘

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

匹丐月井日﹞ $\vec{p}_1\perp\vec{p}_3$ , $\vec{p}_2\perp\vec{p}_3$ 午卅月‘ $\vec{p}_1$ , $\vec{p}_2$ 毛弘仿丞’扑亙立永玄及躂跟祭芊匹懇筋躂跟祭仄﹞ $\vec{p}_3$ 反懇筋祭允木壬方中‘

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

午云仁午﹞ $\left({\vec{q}_i}\,,\,{\vec{q}_j}\right)=\delta_{ij}$ () 互嶽曰峏切懇筋躂跟煙午卅月‘ 動曉方曰墊昫反

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

午躂跟墊昫匹覆剩祭今木月‘

鏜啦 5.75 (覆憊墊昫及裔肣塢棉) 瞬溥庍晶 $f:\mathbb{R}^3\to\mathbb{R}^3$ ; $\vec{y}=A\vec{x}$ 及裔肣塢棉反

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

匹丐月‘ $\dim(W(2;f))=2$ , $\dim(W(-4;f))=1$ , $W(2;f)\cap W(-4;f)=\{\vec{0}\}$ 方曰

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

毛丿月‘ 裔肣塢棉 , 反 $\mathbb{R}^3$ 及躂狟坌荸匹丐月‘ 引凶﹞ 唱卅月裔肣襖卞簞允月裔肣矛弁玄伙反躂跟允月及匹﹞ 裔肣塢棉手躂跟仄 $W(2;f)\perp W(-4;f)$ 毛丿月‘ $W(2;f)\perp W(-4;f)$ 午(↗)方曰﹞ 反 $\mathbb{R}^3$ 卞云仃月及躂跟怀塢棉午卅月‘ 引凶菅卞反 $\mathbb{R}^3$ 卞云仃月及躂跟怀塢棉午卅月‘