The claimed result is an immediate consequence of the following more informative characterization.

THEOREM 1. Let P be an idempotent matrix; i.e., let P² = P. Then P⁺ is idempotent if and only if P = P*, with P⁺ and P* denoting the Moore-Penrose inverse and the conjugate transpose of P, respectively.

Proof. Let P² = P = P*. Then P = P³ = (P²)*, which in turn directly implies that the Penrose defining equations of P⁺ are satisfied for P⁺ = P, and so our proof of sufficiency is complete. For proving necessity, let P be such that P² = P and (P⁺)² = P⁺. If P = 0, then trivially P⁺ = 0 = P*. Next, let P ≠ 0 and let r denote the rank of P. According to the well-known singular value decomposition theorem, P can then be written as P = UD_rV*, where U and V are (column-unitary matrices) such that U*U = V*V = I_r and where D_r is an r × r diagonal matrix with the r positive singular values of P along its main diagonal. In view of this decomposition, clearly

Because P⁺ = V(D_r)⁻¹U*, which again is easily seen by checking the Penrose defining equations of P⁺, we likewise obtain

Combining these observations necessarily results in D_r = I = U*V = V*U. Consequently, (V − U)*(V − U) = 0, or, equivalently, V = U. As claimed, we thus arrive at P = UU* = P*, and so our proof is complete. █

Further equivalent conditions for an idempotent matrix P to be Hermitian are given in the following theorem.

THEOREM 2. Let P be an idempotent matrix. The following conditions are then equivalent:

(c)

; i.e., P is an EP matrix.

That (a) [hArr ] (b) [hArr ] (c) has been shown recently in Werner (2002). That (a) [hArr ] (e) [hArr ] (f) can be shown on similar lines and is therefore left to the reader.