Prove that P = A†A is orthogonal projection where A† is pseudo-inverse of A

Question

I am new to this but I was wondering how to prove this. I can reduce A†A to (VΣ†UT)(UΣVT). Would I have to reduce the UT and U to the identity matrix and just continue to simplify? I would get something like (VΣT)...(ΣVT), I guess I need help doing a full simplification of A†A

Answer 1

To start with, as you say, $U^TU$ is the identity. That gives you $$A^\dagger A=V\Sigma^\dagger\Sigma V^T$$ Now, $\Pi=\Sigma^\dagger\Sigma$ is a diagonal matrix with 0s and 1s on the diagonal, so $V\Pi V^T$ is the eigendecomposition of a projection matrix.