Find jordan from of matrix

Question

I' am trying to find Jordan form of given matrix: \begin{bmatrix} 1 & 2 & 0 \\ -1 & -1 & -1 \\ 0 & 0 & 1 \end{bmatrix} So far i founD characteristic polynomial : $(1 - \lambda )[(1-\lambda)(-1-\lambda)+2]$, eingenvalue: $\lambda = 1$ and eigenvector: $(1,0,-1)$. Now i need to find basis for matrix but i don't know how to do this if there is only one eigenvector.

Answer 1

The roots of the characteristic polynomial are $1$, $i$, and $-i$. So, the Jordan normal form of your matrix is the diagonal matrix$$\begin{bmatrix}1&0&0\\0&i&0\\0&0&-i\end{bmatrix}.$$