I'm preparing for a test and somewhat noticed I don't understand everything as well as I thought I did.

I'm trying to solve the following problem:

Let

be the projection on the plane

(a) Give a basis of

consisting of eigenvectors of

Since

for all v in V, I guess we can take two orthogonal vectors

. But how do we get a third eigenvector? Just taking a third orthogonal vector?

(b) Give the matrix of

with respect to the standard-basis in

(c) Is

normal?

So since

is real I should figure out whether

. I guess I should find

first.

Any help is appreciated.