1. Recall that a square matrix A is called Hermition if A = A*. A Hermitian matrix A e M, (C) is called positive definite if it has the property that *Ar > 0 for every non-zero vector r in C". a) Prove that if A E M, (C) is Hermitian, then A is positive definite if and only if it can be written as UDU* where U is unitary and D is a diagonal matrix with all diagonal entries positive. b) Prove that A E Mn (C) is positive definite if and only if it can be written as A = YY* for some invertible matrix Y