# Inverse of orthogonal projection

I have an $N \times N$ orthogonal projection matrix $P = A^H(AA^H)^{-1}A$ that I'm trying to find the inverse for. I'm using matlab, however, I keep getting the warning "the matrix is close to singular or badly scaled". Now I'm wondering if it's even invertible.

Are orthogonal projection matrices invertible and if so, is there any stable method for computing the inverse?

-
using $A$ and $A^{\dagger}$ (psuedo-inverse) appropriately will give you the projection matrix – dineshdileep Feb 22 '13 at 19:01

This is because projection matrices satisfy $P^2 = P$ or $P(P - I) = 0$. If $P$ is invertible then this implies $P - I = 0$ or $P = I$.