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

Problem:

Write \(A = P C P^{-1}\) for the matrix 
\[A = \begin{bmatrix} 
-8 & -5   \\ 
10 & 2
\end{bmatrix},\]
with \(C\) a rotation matrix corresponding to a complex eigenvalue \(\lambda = a + b i\).

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