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?
Best Answer
If a projection matrix is invertible then it is the identity matrix.
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 $P = I$.
Another way of seeing that projections are usually not invertible is to think about their nullspace. If you are projecting onto a space of smaller dimension then the nullspace is not zero and hence the matrix is not invertible.