This is my first time solving a problem like this and I just wanted to make sure if what I did was correct.
\begin{equation} x'_1=5x_1 + 2 x_2 - x_3 \\ x'_2=-2x_1 + x_2 - 2x_3 \\ x'_3=-6x_1 - 6 x_2 \end{equation}
Using the eigenvalues and eigenvectors, I found the matrix $P$ such that $P^{-1} AP$ is a diagonal matrix where $A$ is the coefficient matrix of the system above.
$P=$ \begin{pmatrix} 0&-1&-1\\ 1&1&0\\ 2&0&1 \end{pmatrix}
$P^{-1} AP=$ \begin{pmatrix} -3&0&0\\ 0&3&0\\ 0&0&6 \end{pmatrix}
I am sure everything above is correct. So assuming it is correct, is the following process correct?
$U'=(P^{-1} AP)U$
\begin{equation} u'_1=-3u_1 \\ u'_2=3u_2 \\ u'_3=6u_3 \end{equation}
\begin{equation} u_1=c_1e^{-3x} \\ u_2=c_2e^{3x} \\ u_3=c_3e^{6x} \end{equation}
$U=$ \begin{pmatrix} c_1e^{-3x} \\ c_2e^{3x} \\ c_3e^{6x} \end{pmatrix}
$X=PU$
I understand this is pretty long. I'd appreciate if someone can just give a quick look at the process and the substitutions I made (with U and U'). Thanks.
EDIT Comment below helped me answer my own question.