Matrix theory Question#4.

Q1:

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 ( using the standard inner product on ).

b) For an arbitrary p×q matrix A, and an arbitrary vector x , show that ( )x is the orthogonal projection of x onto the null space of A. That is, prove that P=) satisfies the conditions in (a), and also prove that the image space of p is the entire null space of A.

c) es P

2 = P, P is called the

orthogonal projection onto its image space.