\[\lambda = -5 - 3i\] \[C = \begin{bmatrix} -5 & -3 \\ 3 & -5 \end{bmatrix}\] \[A = \begin{bmatrix} -1 & 0 \\ -1 & -1 \end{bmatrix} \begin{bmatrix} -5 & -3 \\ 3 & -5 \end{bmatrix} \begin{bmatrix} -1 & 0 \\ 1 & -1 \end{bmatrix}\]

Problem:

Write \(A = P C P^{-1}\) for the matrix 
\[A = \begin{bmatrix} 
-2 & -3   \\ 
6 & -8
\end{bmatrix},\]
with \(C\) a rotation matrix corresponding to a complex eigenvalue \(\lambda = a + b i\).

\begin{solution}
Solution:
\[\lambda = -5 - 3i\]
\[C = \begin{bmatrix} 
-5 & -3   \\ 
3 & -5
\end{bmatrix}\]
\[A = \begin{bmatrix} 
-1 & 0   \\ 
-1 & -1
\end{bmatrix} \begin{bmatrix} 
-5 & -3   \\ 
3 & -5
\end{bmatrix} \begin{bmatrix} 
-1 & 0   \\ 
1 & -1
\end{bmatrix}\]
\end{solution}