Let P in

be a projector with P^H its conjugate transpose.

(a) P^HP is idempotent ⇒ P is Hermitian.

By contradiction. Let P be an oblique projector of rank r. Choose an orthonormal set of r (resp. s = n − r) eigenvectors with eigenvalue 1 (resp. 0). These n eigenvectors form a basis. The matrix C that collects these eigenvectors as its columns satisfies

with J_r the diagonal matrix with on the diagonal r ones and n − r zeros (the “ones” are at the top left); and with I_r (resp. I_s) the identity matrix of dimension r (resp. s). The matrix

and is nonzero (as the projector P is oblique). In this notation we have

Now, the matrix P^HP is idempotent if and only if the matrix D = C⁻¹P^HPC is idempotent. Note that D = C⁻¹(C⁻¹)^HJ_r C⁻¹C = C⁻¹(C⁻¹)^HJ_r = (C^HC)⁻¹J_r. Hence, D is of rank r and is of the form

with

satisfying

Thus, E is nonsingular. Now, suppose that D is idempotent. Then, E is idempotent. Hence, E = I_r and TT^H = 0. This conflicts with T being nonzero. Conclude that D, and therefore also P^HP, is not idempotent.

(a) P is Hermitian ⇒ P^HP is idempotent.

If P = P^H, then the composition P^HP coincides with P and is idempotent.

(b) P is Hermitian [hArr ] PP^H is idempotent.

The projector P is Hermitian if and only if the projector P^H is Hermitian. Apply condition (a) upon P^H and obtain that the projector P is Hermitian if and only if the composition (P^H)^H(P^H) = PP^H is idempotent.