Hi there. I must give the eigenvalues and the eigenvectors for the matrix transformation of the orthogonal projection over the plane XY on $\displaystyle R^3$

So, at first I thought it should be the eigenvalue 1, and the eigenvectors (1,0,0) and (0,1,0), because they don't change. But I also tried doing the calculus, and then I've confused.

I have:

$\displaystyle [T]_c=\begin{bmatrix}{1}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0 }&{0}\end{bmatrix}$

From the characteristic polynomial I get to: $\displaystyle -\lambda(1-\lambda)^2$

Then, I have as eigenvalues 0, and 1 twice.

$\displaystyle \lambda_1=0$:

I get to: $\displaystyle \begin{Bmatrix}x=0\\y=0\end{matrix}$

and then the eigenvector: $\displaystyle {(0,0,1)}$

So I thought, shouldn't it be zero? because of the projection. I have doubts with this, but I know that as it is a symmetrical matrix it should be diagonalizable, and then I should get a basis from the eigenvectors, which I wouldn't find with just the first reasoning, and then I need a linear independent vector, like this one, respect to the first I gave.

And then for $\displaystyle \lambda_2,\lambda_3=1$:

z=0,

Which gives: $\displaystyle {(x,y,0)}$, and implies: $\displaystyle {(1,0,0),(0,1,0)}$, I think that have sense.

Well, I need some help with this. Can anybody tell me if this is right, and if it isn't, what I did wrong?

Thank you!

Bye there.

PS: I have my final exam tomorrow :P