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

Problem:

Write \(A = P C P^{-1}\) for the matrix 
\[A = \begin{bmatrix} 
-12 & 5   \\ 
-13 & 4
\end{bmatrix},\]
with \(C\) a rotation matrix corresponding to a complex eigenvalue \(\lambda = a + b i\).

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