Special Gaussian family on \(\mathbb{C}^{3}\) : Let \(ho=\left(ho_{1}, ho_{2}, ho_{3}ight)\) be a real vector, and let \(Z=\left(Z_{1}, Z_{2}, Z_{3}ight)\) be a zero-mean complex Gaussian variable with covariance matrix of the form

\[
\operatorname{cov}\left(Z, Z^{*}ight)=\left(\begin{array}{ccc}
1 & -i ho_{3} & i ho_{2} \\
i ho_{3} & 1 & -i ho_{1} \\
-i ho_{2} & i ho_{1} & 1
\end{array}ight)=I_{3}+i \chi(ho)
\]

Show that the covariance matrix is positive definite if and only if \(\|ho\| \leq 1\). For any real \(3 \times 3\) orthogonal matrix \(L\) with \(\operatorname{det}(L)=1\), show that \(L Z\) belongs to the same family with parameter \(L ho\), i.e., that \(\chi(L ho)=L \chi(ho) L^{\prime}\).