I'm stuck on this problem and I need help, any kind of help would do.
So i'm supposed to find a basis for $R^3$ which consists of the eigenvectors for $A$ and a matrix $M$ so that
$M^{-1}AM=\begin{bmatrix}\lambda_{1}&0&0\\0&\lambda_{2}&0\\0&0&\lambda_{3}\end{bmatrix}$
I'm know that $A=\begin{bmatrix}0&9&0\\1&0&0\\0&1&0\end{bmatrix}$
I've found out that the eigenvalues and eigenvectors of the matrix $A$ is $\lambda_{values}={-3, 0 ,3}$ which gives us that the eigenvectors is equals to $\lambda_{vectors}=v_{1}=(0, 0, 1), v_{2}=(9, 3, 1) , v_{3}=(9, -3,1)$
which should give us the matrix $M=\begin{bmatrix}0&9&9\\0&3&-3\\1&1&1\end{bmatrix}$
and this should give us that $$M^{-1}=\begin{bmatrix}-\frac{1}{9}&0&1\\\frac{1}{18}&\frac{1}{6}&0\\\frac{1}{18}&-\frac{1}{6}&0\end{bmatrix}$$
this is where I knew i've done something wrong, since I cannot get
$\begin{bmatrix}\lambda_{1}&0&0\\0&\lambda_{2}&0\\0&0&\lambda_{3}\end{bmatrix}$
with the values of $M$, $A$ and $M^{-1}$
Perhaps that you computed $M.A.M^{-1}$. You should have computed $M^{-1}.A.M$.