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 suppressed.

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 = (X0

c, X0 s)0 has a multivariate normal distribution with density p(Z) = (2π)

−p|Σ|

−1/2 exp

−1 2

(Z − µ)

0

Σ−1(Z − µ)



, where µ = (M0 c,M0 s)0 is the mean vector. Prove

|Σ| =

1 2

2p

|C − iQ|

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

(v0

c, v0 s)0 and (v0 s, −v0 c)0

, where vc − ivs denotes the eigenvector of fxx. Show that