Exercise 1.6.15

• 1.

  $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 3\hfill \end{array}\right]$, it has eigenvalues ${\lambda }_1=1$, and ${\lambda }_2=3$. It has eigenvectors $v_1=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \end{array}\right]$, $v_2=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right]$. So we have $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 3\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 1\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

$A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 3\hfill \end{array}\right]$, it has eigenvalues ${\lambda }_1=0$, and ${\lambda }_2=4$. It has eigenvectors $v_1=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right]$, $v_2=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 3\hfill \end{array}\right]$. So we have $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 3\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 4\hfill \end{array}\right]\left[\begin{array}{cc}\hfill \frac{3}{4}\hfill & \hfill -\frac{1}{4}\hfill \\ \hfill \frac{1}{4}\hfill & \hfill \frac{1}{4}\hfill \end{array}\right]$

• 1.

  If $A=\mathit{X\Lambda }{X}^{-1}$, then $A^3=X{\Lambda }^3{X}^{-1}$ and $A^{-1}=X{\Lambda }^{-1}{X}^{-1}$