$\rm det(\left[\begin{array}{ccc} 3&4&0\\4&3&0\\0&0&7 \end{array}\right]-λ\left[\begin{array}{ccc} 3&4&0\\4&3&0\\0&0&7 \end{array}\right])=det\left[\begin{array}{ccc} 3-λ&4&0\\4&3-λ&0\\0&0&7-λ \end{array}\right]$

$\rm =(3-λ).det\left[\begin{array}{ccc} 3-λ&0\\0&7-λ \end{array}\right]-4.det\left[\begin{array}{ccc} 4&0\\0&7-λ \end{array}\right]$

$\rm =(3-λ)(λ^2-10λ+21)-4.4(-λ+7)=-λ^3+13λ^2-35λ-49$

`-` Giải:

$\rm -λ^3+13λ^2-35λ-49=0$

`<=>`$\rm -(λ+1)(λ-7)^2=0$

`=>`$\rm λ=-1,λ=7$

`-` Tìm vecto riêng.

$\rm (A-λI)=\left[\begin{array}{ccc} 3&4&0\\4&3&0\\0&0&7 \end{array}\right]-(-1)\left[\begin{array}{ccc} 1&0&0\\0&1&0\\0&0&1 \end{array}\right]$

$\rm =\left[\begin{array}{ccc} 4&4&0\\4&4&0\\0&0&8 \end{array}\right]$

$\rm \left[\begin{array}{ccc} 4&4&0\\4&4&0\\0&0&8 \end{array}\right] \mathop{\to}\limits^{\dfrac{1}{4}h_1\to h_1}\left[\begin{array}{ccc} 1&1&0\\4&4&0\\0&0&8 \end{array}\right] $

$\rm \left[\begin{array}{ccc} 1&1&0\\4&4&0\\0&0&8 \end{array}\right] \mathop{\to}\limits^{h_2-4h_1\to h_1}\left[\begin{array}{ccc} 1&1&0\\0&0&0\\0&0&8 \end{array}\right] $

$\rm \left[\begin{array}{ccc} 1&1&0\\0&0&0\\0&0&8 \end{array}\right] \mathop{\to}\limits^{h_2⇔h_3}\left[\begin{array}{ccc} 1&1&0\\0&0&8\\0&0&0 \end{array}\right] $

$\rm \left[\begin{array}{ccc} 1&1&0\\0&0&8\\0&0&0 \end{array}\right] \mathop{\to}\limits^{\dfrac{1}{8}h_2\to h_2}\left[\begin{array}{ccc} 1&1&0\\0&0&1\\0&0&0 \end{array}\right] $

`-` 

$\rm (A+λI) \left[\begin{array}{ccc} x\\y\\z \end{array}\right] =\left[\begin{array}{ccc} 1&1&0\\0&0&1\\0&0&0 \end{array}\right]\left[\begin{array}{ccc} x\\y\\z \end{array}\right]=\left[\begin{array}{ccc} 0\\0\\0 \end{array}\right]$

`=>`$\begin{cases} x+y=0\\z=0 \end{cases}$`=>`$\begin{cases} x=-y\\z=0 \end{cases}$

`-` Thay vào $\rm \left[\begin{array}{ccc} x\\y\\z \end{array}\right]$

$\rm η=\left[\begin{array}{ccc} -y\\y\\0 \end{array}\right]$

`-` Chọn $\rm y=1$

`=>`$\rm \left[\begin{array}{ccc} -1\\1\\0 \end{array}\right]$

$\rm \bullet$ Tương tự cho: $\rm λ=7$ sẽ thu được $\rm \left[\begin{array}{ccc} 0\\0\\1 \end{array}\right]$ và $\rm \left[\begin{array}{ccc} 1\\1\\0 \end{array}\right]$