# Whitening on projection matrices

The projection matrix $$P = I -xx^T\in \mathbf{R}^{d \times d}$$ has a zero eigenvalue and eigenvalues equal to one with multiplicity $$d-1$$. Is it possible to apply whitening transform on $$P$$ taking into consideration that the inverse of $$D=diag(0,1,1,\dots, 1)$$ has to be computed? Is there any way to overcome this difficulty?