By writing the complex vector \(z\) and the Hermitian matrix \(\Gamma\) as a linear combination of real and imaginary parts, show that the Hermitian quadratic form \(z^{*} \Gamma z\) reduces to the following linear combination of real quadratic forms:

\(\left(x^{\prime}-i y^{\prime}ight) \Gamma_{0}(x+i y)+i\left(x^{\prime}-i y^{\prime}ight) \Gamma_{1}(x+i y)=x^{\prime} \Gamma_{0} x+y^{\prime} \Gamma_{0} y+y^{\prime} \Gamma_{1} x-x^{\prime} \Gamma_{1} y\).

Hence deduce that the real and imaginary parts of \(Z \sim \mathbb{C N}(0, \Sigma)\) are identically distributed Gaussian vectors \(N\left(0, \Sigma_{0}ight)\) with covariances \(\operatorname{cov}(X, Y)=\) \(-\operatorname{cov}(Y, X)=\Sigma_{1}\).

Gaussian Linear Prediction The next five exercises are concerned with estimation and prediction in the Gaussian linear model \(Y \sim N_{n}\left(\mu=X \beta, \sigma^{2} Vight)\) in which the observation is the linear transformation \(Z=K Y\). The matrices \(X\) of order \(n \times p\), \(K\) of order \(n-k \times n\), and \(V\) of order \(n \times n\) are given, while \(\beta, \sigma^{2}\) are parameters to be estimated. All three matrices are of full rank, the product \(K X\) has rank \(p \leq n-k\), while the Hilbert space \(\mathcal{H}\) with inner-product matrix \(W=V^{-1}\) determines the geometry.