# calculate triangular form of a matrix

The following I don't understand. I would really appreciate any help! I want to find a flag basis for the following matrix:

$A=\begin{bmatrix} 3&0&-2\\ -2&0&1\\ 2&1&0 \end{bmatrix}$

I calculated the eigenvector, which is: $v=\begin{bmatrix}1\\-1\\1\end{bmatrix}$. Then I extend v to basis:$B=\{v,e_{2},e_{3}\}$ and calculated the matrix with respect to the new basis: $\begin{bmatrix}1&0&-2\\0&0&-1\\0&1&2\end{bmatrix}$. The next step I do not understand. Why is it okay to look only at the follwong matrix and calucalte eigenvectors etc. : $\begin{bmatrix}0&-1\\1&2\end{bmatrix}$

As far as I know this matrix represents a certain endomorphism $\phi$ that is invariant on a certain subspace.But why is it not okay to look at $\begin{bmatrix}0&0&0\\0&0&-1\\0&1&2\end{bmatrix}$? As far as I understand it, if you calculate the eigenvectors of the last matrix you get the same as if you calculated the eigenvectors of the 2x2 matrix plus adding a zero in the right spot. This matrix seems to me somehow more "appropiate". Furthermore I dont understand way I get the same eigenvectors for the matrix $\begin{bmatrix}1&0&0\\0&0&-1\\0&1&2\end{bmatrix}$. Does this matrix also represent $\phi$? Thank you very much for you help in advance!