Let X and Z be statistically independent Gaussian rv s of arbitrary dimension n and m respectively. Let Y = [H]X + Z, where [H] is an arbitrary real n × m matrix.
(a) Explain why X1, ... , Xn, Z1, ... , Zm must be jointly Gaussian rv s. Then explain why X1, ... , Xn, Y1, ... , Ym must be jointly Gaussian.
(b) Show that if [KX ] and [KZ] are non-singular, then the combined covariance matrix [K] for (X1, ... , Xn, Y1, ... , Ym)T must be non-singular.