TITLE: How to diagonalize this matrix...
QUESTION [28 upvotes]: Can someone show me step-by-step how to diagonalize this matrix? I'm trying to teach myself differential equations + linear algebra, but I'm stumped on how to do this. I'd really appreciate if someone would take the time to do this with me! 
$\begin{bmatrix}
1 & 2 & 0 \\
2 & 1 & 0 \\
0 & 0 & -3 \\
\end{bmatrix}$

REPLY [4 votes]: Since Bye_World has done everything, I just want to try proving his statement:

This equation, $(A−\lambda I)x=0$ has a nontrivial solution (a solution where $x\neq 0$) if and only if $\det(A−\lambda I)=0$"

We first assume that $B = A−\lambda I$ is invertible (or $\det(B)≠0$). If this is true, the equation can be rewritten as:
$$B^{-1}Bx = B^{-1}0=0$$
Since $B^{-1}B=I$ (the identify matrix), then $x=0$, which is false.
Then B is not invertible or $\det(B)=0$.