\[\lambda = 4 - 2i\] \[C = \begin{bmatrix} 4 & -2 \\ 2 & 4 \end{bmatrix}\] \[A = \begin{bmatrix} -1 & 1 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} 4 & -2 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} -1 & -1 \\ 0 & -1 \end{bmatrix}\]

Problem:

Write \(A = P C P^{-1}\) for the matrix 
\[A = \begin{bmatrix} 
2 & -4   \\ 
2 & 6
\end{bmatrix},\]
with \(C\) a rotation matrix corresponding to a complex eigenvalue \(\lambda = a + b i\).

\begin{solution}
Solution:
\[\lambda = 4 - 2i\]
\[C = \begin{bmatrix} 
4 & -2   \\ 
2 & 4
\end{bmatrix}\]
\[A = \begin{bmatrix} 
-1 & 1   \\ 
0 & -1
\end{bmatrix} \begin{bmatrix} 
4 & -2   \\ 
2 & 4
\end{bmatrix} \begin{bmatrix} 
-1 & -1   \\ 
0 & -1
\end{bmatrix}\]
\end{solution}