\[\lambda = 4 + 2i\] \[C = \begin{bmatrix} 4 & 2 \\ -2 & 4 \end{bmatrix}\] \[A = \begin{bmatrix} 4 & -1 \\ 3 & -1 \end{bmatrix} \begin{bmatrix} 4 & 2 \\ -2 & 4 \end{bmatrix} \begin{bmatrix} 1 & -1 \\ 3 & -4 \end{bmatrix}\]

Problem:

Write \(A = P C P^{-1}\) for the matrix 
\[A = \begin{bmatrix} 
30 & -34   \\ 
20 & -22
\end{bmatrix},\]
with \(C\) a rotation matrix corresponding to a complex eigenvalue \(\lambda = a + b i\).

\begin{solution}
Solution:
\[\lambda = 4 + 2i\]
\[C = \begin{bmatrix} 
4 & 2   \\ 
-2 & 4
\end{bmatrix}\]
\[A = \begin{bmatrix} 
4 & -1   \\ 
3 & -1
\end{bmatrix} \begin{bmatrix} 
4 & 2   \\ 
-2 & 4
\end{bmatrix} \begin{bmatrix} 
1 & -1   \\ 
3 & -4
\end{bmatrix}\]
\end{solution}