Orthogonal projection on vector subspaces arises in many applied field
s. The common assumption about the orthogonal complementary subspace i
s that it is spanned by white noise components. We generalize a previo
us perturbation analysis of projection operators to that with a noise
field with an arbitrarily structured covariance matrix. The resulting
expressions are insightful, and their algebraic power is very useful f
or applications.