Jordan Normal Form of an Orthogonal Projection

Question

I have the following question in my exam preparation:

I have to:

Find the minimal polynomial of T and the Jordan Normal Form of T.

What I can't understand is what to do with the fact of the projection.

How to find a Jordan Normal Form I know - find the characteristic, then the polynomial minimal polynomial.

But first I have to find a basis to work with, and I can't understand how to do that.

Thanks,

Alan

Answer 1

Since $T$ is a non-zero orthogonal projection one has $T^2=T$ which means that the minimal polynomial is $p(z)=z^2-z$. For the orthonormal basis: take $e_1=(\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}})$ (a vector of norm $1$ which spans the range of $T$) and two vectors which are of norm $1$ and are orthogonal to $e_1$ (they span the kernel of $T$). One possibility is $e_2=(0, 1, 0)$ and $e_3=(\frac{1}{\sqrt{2}}, 0, -\frac{1}{\sqrt{2}})$.