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## Homework Statement

Hi there. I must give the eigenvalues and the eigenvectors for the matrix transformation of the orthogonal projection over the plane XY on [tex]R^3[/tex]

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:

[tex][T]_c=\begin{bmatrix}{1}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{0}\end{bmatrix}[/tex]

From the characteristic polynomial I get to: [tex]-\lambda(1-\lambda)^2[/tex]

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

[tex]\lambda_1=0[/tex]:

I get to: [tex]\begin{Bmatrix}x=0\\y=0\end{matrix}[/tex]

and then the eigenvector: [tex]{(0,0,1)}[/tex]

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 [tex]\lambda_2,\lambda_3=1[/tex]:

z=0,

Which gives: [tex]{(x,y,0)}[/tex], and implies: [tex]{(1,0,0),(0,1,0)}[/tex], 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

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