How to find the eigenvectors of this $3 \times 3$ matrix?

Question

I need to find eigenvectors of the following matrix?

\begin{bmatrix}2&6&-15\\1&1&-5\\1&2&-6\end{bmatrix}

What I have done: I found eigenvalue $\lambda = -1$, then

\begin{bmatrix}3&6&-15\\1&2&-5\\1&2&-5\end{bmatrix}

\begin{bmatrix}1\;&2\;&-5\:\end{bmatrix}

$$x_1 = \begin{bmatrix}3\\ 1\\ 1\end{bmatrix}\qquad x_2 = \begin{bmatrix}-2\\1\\0\end{bmatrix}$$ How to find the last one?

Answer 1

As I indicated in comment, take

$$ P = \left( \begin{array}{rrr} -2 & 3 & 1 \\ 1&1&0 \\ 0&1&0 \end{array} \right) $$ then calculate $P^{-1}$ and finally $ J=P^{-1} AP,$ where $A$ is your original matrix. I got lucky, as this $\det P = 1.$

The reason it is important to practice finding such a $P$ is that most applications use the reverse order, namely expressing $A = PJ P^{-1}$