Matrix theory Question#4.

a) Recall that if a p× p matrix P is Hermitian and satisfies P² =P, P is called the orthogonal projection onto its image space. Prove that for such a matrix P, Px is orthogonal to ( x – Px ) for any vector x∈C^p ( using the standard inner product on C^p ).
b) For an arbitrary p×q matrix A, and an arbitrary vector x∈ R^q , show that ( I_q- A^+ A)x is the orthogonal projection of x onto the null space of A. That is, prove that P=(I_q- A^+ A) satisfies the conditions in (a), and also prove that the image space of p is the entire null space of A.