Let A C"X". Show that there exists a unitary matrix Q such that A = QAQ. Problem 8 Let A CMX" has rank n. Show that there is a factorization of A = PH, where PE CX has orthonormal columns, and He C**" is Hermitian positive definite. If A C" *", then, show that, JA- Pl2 S||A - Ql2 for any unitary matrix Q. Problem 9 Let A E CXN has rank n have a factorization A PH as in Problem 8. Show that |A' A - I12 0. Show that o is a singular values of A if and only if the matrix 1. A -ol A* is singular.