Let $\phi$ a linear application and $A$ his matrix such that $$A=\begin{pmatrix}2&2&-3\\ 5&1&-5\\ -3&4&0\end{pmatrix}$$
I want to find a inversible matrix $P$ such that there exist a Upper Triangular matrix $D$ with the equation:
$$A=P^{-1} D P$$
This is an exemple that has been built by me to motivate my question. Because this matrix $A$ has only one eigenvector which is $v=(1,1,1)$. That means $A$ is not diagonalizable.
Question 1: How can i find a basis of $P$?
Question 2. How many matrix $D$ are there?