7.1 Consider the complex Gaussian distribution for the random variable X = Xc iXs, as defined in (7.1)-(7.3), where the argument k has been

7.1 Consider the complex Gaussian distribution for the random variable X =

Xc − iXs, as defined in (7.1)-(7.3), where the argument ωk has been suppressed. Now, the 2p × 1 real random variable Z = (X

c,X

s) has a multivariate normal distribution with density

using the result that the eigenvectors and eigenvalues of Σ occur in pairs, i.e., (v

c, v
s) and (v
s,−v
c), where vc−ivs denotes the eigenvector of fxx.
Show that

so p(X) = p(Z) and we can identify the density of the complex multivariate normal variable X with that of the real multivariate normal Z.

P(Z) (2) 1/2 == (Z- where (MM)' is the mean vector. Prove = 2p - (+) |C-iQ|,