1 1 1 20. x' = %3D 2 1 2 N O O

given

$x^{\text{'}}=\left[\begin{array}{cccc}2& 1& 0& 1\\ 0& 2& 1& 0\\ 0& 0& 2& 1\\ 0& 0& 0& 2\end{array}\right]x$

we will find the eigenvalue 

$$\begin{gathered} A-\lambda I=\left[\begin{array}{cccc}2& 1& 0& 1\\ 0& 2& 1& 0\\ 0& 0& 2& 1\\ 0& 0& 0& 2\end{array}\right]-\lambda \left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right] \\ =\left[\begin{array}{cccc}2-\lambda & 1& 0& 1\\ 0& 2-\lambda & 1& 0\\ 0& 0& 2-\lambda & 1\\ 0& 0& 0& 2-\lambda \end{array}\right] \end{gathered}$$

$$\begin{gathered} \left|A-\lambda I\right|=\left|\begin{array}{cccc}2-\lambda & 1& 0& 1\\ 0& 2-\lambda & 1& 0\\ 0& 0& 2-\lambda & 1\\ 0& 0& 0& 2-\lambda \end{array}\right| \\ =\left(2-\lambda \right)\left[\left(2-\lambda \right)\left(2-\lambda \right)\right]-0+0+0 \\ ={\left(2-\lambda \right)}^3 \end{gathered}$$

$$\begin{gathered} {\left(2-\lambda \right)}^3=0 \\ \Rightarrow \lambda =2,2,2 \end{gathered}$$

implies 2 is the only eigenvalue with multiplicity 3

now, we will find the eigenvector for $\lambda =2$

$$\begin{gathered} A-2I=\left[\begin{array}{cccc}0& 1& 0& 1\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=0 \\ \Rightarrow x_2+x_4=0 \\ \Rightarrow x_3=0 \\ \Rightarrow x_4=0 \\ \Rightarrow x_2=0 \end{gathered}$$

implies we have a eigenvector $v_4=\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right]$

now, we will find other vectors

$\left[\begin{array}{cccc}2& 1& 0& 1\\ 0& 2& 1& 0\\ 0& 0& 2& 1\\ 0& 0& 0& 2\end{array}\right]\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right]=\left[\begin{array}{c}2\\ 0\\ 0\\ 0\end{array}\right]=v_3$

$\left[\begin{array}{cccc}2& 1& 0& 1\\ 0& 2& 1& 0\\ 0& 0& 2& 1\\ 0& 0& 0& 2\end{array}\right]\left[\begin{array}{c}2\\ 0\\ 0\\ 0\end{array}\right]=\left[\begin{array}{c}4\\ 0\\ 0\\ 0\end{array}\right]=v_2\begin{array}{}\\ \\ \\ \end{array}$

$\left[\begin{array}{cccc}2& 1& 0& 1\\ 0& 2& 1& 0\\ 0& 0& 2& 1\\ 0& 0& 0& 2\end{array}\right]\left[\begin{array}{c}4\\ 0\\ 0\\ 0\end{array}\right]=\left[\begin{array}{c}8\\ 0\\ 0\\ 0\end{array}\right]=v_1$