I don't understand waht to do for diagonalization?

OK so on my homework it says Diagonalize the following matrix, if possible:

[3 2 -2]

[0 2 0]

[0 1 3]

I found out the eigenvalues which is 2 and 3. 3 is a repetition so that's why there's only 2.

Now I'm suppose to find the eigenvalue. But I have no clue how to find the eigenvector. I'm readingm y textbook and the only thing that happens is they give me the eigenvalue then mysteriously find the eigenvector within one step.

Re: I don't understand waht to do for diagonalization?

If all else fails (or maybe it should cross your mind as an earlier choice), resort to the definitions.

Your 3x3 matrix - A

Your eigenvalue - 3

An arbitrary column vector -

Ax = 3x

Solve.

Re: I don't understand waht to do for diagonalization?

So I did Ax - 3x=0 and I got a row reduced 3x3 matrix with 2's as the diagonals Everything else are 0s... Is this suppose to be right?

I'm sorry, but I'm just really bad at this stuff >.>. I'm not even sure if I can pass the class.