On the spectrum and spectrum multiplicities of a sum of orthogonal projections - Kyrychenko, A.A.; Samoılenko, Yu.S. (Algebra and Discrete Mathematics, 2004)
Let H be a unitary (finite dimensional Hilbert) space. It is known [1] that a self-adjoint operator on H, A = A* ≥ 0, is a sum of n orthogonal projections for some n ∈ N if and only if 1) tr A ∈ N∪{0}, 2) tr A ≥ dim Im A (Im A = AH)