ABSTRACT
W=CAC*, and From (5.71) W is distributed as the complex Wishart Wc(n, I) with n=N-1 degrees of freedom and parameter I. Let Q be an p×p unitary matrix with first row
and the remaining rows are defined arbitrarily. Writing U=(U1,…, Up)'= QV, B=QWQ* we obtain
where B is partitioned as
where B22 is (p-1)×(p-1). From (5.71) taking S=I the joint pdf of B22, B12 and
is
where
From (6.93), using (5.71) we conclude that H is independent of B22 and B12; and H is distributed as complex Wishart with degrees of freedom N-p and parameter unity. From Theorem 5.3.4, the conditional distribution of B given Q, is that of
where conditionally given Q, Va, a=2,…, N, are independent and each has a p-variate complex normal distribution mean 0 and
covariance I. Hence given Q, is distributed as where Wa, a=1,…, N-p, given Q are independent and each has a single variate complex normal distribution with mean 0 and variance unity. From Theorem 6.11.1 and the fact that the sum of independent chi-squares is a chisquare we conclude that
conditionally given Q, is distributed as Since this distribution does not
involve is unconditionally distributed as The
quantity 2V*V (using Theorem 6.11.1) is distributed as Hence from Theorem 5.3.5 we get the Theorem. Q.E.D.
