In this problem we extend the proof of theorem 516 to the

Question

In this problem, we extend the proof of Theorem 5.16 to the case when A is m × n with m _X = 0.

(a) Prove there exists an (n - m) × n matrix  of rank n-m with the property that A' = 0.

(b) Let  =  ^-1_X and define the random vector Use Theorem 5.16 for the case m = n to argue that Y¯ is a Gaussian random vector.

(c) Find the covariance matrix  of . Use the result of Problem 5.7.8 to show that Y and Yˆ are independent Gaussian random vectors

Theorem 5.16

Given an n-dimensional Gaussian random vector X with expected valueµ X and covariance CX, and an m × n matrix A with rank(A) = m,

Is an m-dimensional Gaussian random vector with expected value µ_Y = Aµ_X + b and covariance CY = AC_XA'

Problem 5.7.8

An n-dimensional Gaussian vector W has a block diagonal covariance matrix where CX is m × m, C_Y is (n - m) × (n - m). Show that W can be written in terms of component vectors X and Y in the form such that X and Y are independent Gaussian random vectors