Jordan Canonical Form of simple $2\times 2$ matrix complex

Question

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".

Answer 1

Since we have 2 distinct eigenvalues the matrix can be diagonalized and therefore the Jordan form is the diagonal matrix

$$\begin{pmatrix} 1-i & 0 \\ 0 & 1+i \\ \end{pmatrix}$$