In this problem, we extend the proof of Theorem 8.11 to

Chapter 8, Problem 8.50

(choose chapter or problem)

In this problem, we extend the proof of Theorem 8.11 to the case when A is m, x 11, with m < 11,. For this proof, 've assume X is an 11,-dimensional Gaussian vector and that we have proved T heorem 8.11 for the case m = 11,. Since the case m, = ri is sufficient to prove that Y = X + b is Gaussian, it is sufficient to sho'v for m, < 11, that Y = AX is Gaussian in the case 'v hen x = 0. (a) Prov~ there exists an ( 11, - 1n) x 11, matrix A of rank n - m, with the property that AA' = 0 . Hint : Review the Gram- Schmidt procedure. (b) Let A= ACx 1 and define the random vector y = [~] = [1] x Use Theorem 8.11 for t he case m, = n, to argue that Y is a Gaussian random vector. ( c) Find the covariance matrix C of Y. Use the result of 8.5.13 to A show that Y and Y are independent Gaussian random vectors.