# Triangularization of a matrix.

so I need to find an invertible matrix $P$ such that $P^{-1}AP$ is upper-triangular, where $$A = \begin{bmatrix} 4 & -1 \\ 9 & -2 \end{bmatrix}$$

So I found that the eigenvalue is $1$ which lead to an eigenvector of $E = \begin{bmatrix} \frac{1}{3} \\ 1 \end{bmatrix}$. So if I let $v_1= (\frac{1}{3}, 1)$ then I know I am supposed to choose $v_2$ to be some other vector since $P = \begin{bmatrix} v_1 & v_2 \end{bmatrix}$ but I'm not really sure what to pick for $v_2$ or why I should pick it.. Would I just choose $v_2 = (1,0)$?

Choosing $v_2 = (1,0)$ works perfectly well. All you need here is something that is not a multiple of $v_1$ so that $P$ is invertible.
If you want, you can choose $v_1,v_2$ so that $P$ is orthogonal (unitary). In that case, you would be looking for the Schur decomposition of the matrix.