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

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 0

c

, X 0

s

)

0 has a multivariate normal distribution with density p(Z) = (2π)

−p

|Σ|

−1/2 exp

−

1 2

(Z − µ)

0Σ

−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., (v 0

c

, v 0

s

)

0 and (v 0

s

, −v 0

c

)

0

, where vc − ivs denotes the eigenvector of fxx. Show that 1

2

(Z − µ)

0Σ

−1

(Z − µ)) = (X − M)

∗

f

−1

(X − M)

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.