Let
$$A =\begin{pmatrix}
1 & 1 \\
-1 & 1 \\
\end{pmatrix}$$
Original question: determine the Jordan form of this matrix and its corresponding P matrix.
I understand all the steps I need to take to find the Jordan form, but how many times I try I can't seem to do this one the right way..
Using its characteristic polynomial I found that the eigenvalues of $A$ are $λ = 1 \pm i$.
I found the first eigenvector: $$\begin{pmatrix}
i \\
1 \\
\end{pmatrix}$$
Finding a generalized eigenvector is where I go wrong, I think.
I found: $$\begin{pmatrix}
-1 \\
0 \\
\end{pmatrix}$$
Now we can make $P = [v_1\, v_2]$ to find $J$, where $J = (P^{-1})AP$
After computing this the matrix I get for $J$ is not a Jordan form.
The matrix I got is:
$$\begin{pmatrix}
1-i & 1 \\
0 & 1+i \\
\end{pmatrix}$$
Anyone who knows what I did wrong?
Edit: Changed the original question to: "Determine the Jordan form and P matrix" instead of "Determine the Jordan form".
Best Answer
Since we have 2 distinct eigenvalues the matrix can be diagonalized and therefore the Jordan form is the diagonal matrix
$$\begin{pmatrix} 1-i & 0 \\ 0 & 1+i \\ \end{pmatrix}$$