Consider four random complex vectors $\mu_i$ of length $K$ whose entries are drawn from the complex normal distribution $\mathcal{CN}(\mathbf{0},\mathbb{1})$ centered in zero and of unit variance. Then form the $2 \times 2$ complex "normalized" matrix $$ \chi=\frac{1}{\sum_i|\mu_i|^2}\begin{pmatrix}\mu_1^*\\\mu_2^*\end{pmatrix}\cdot\begin{pmatrix}\mu_3&\mu_4\end{pmatrix}=\frac{1}{\sum_i|\mu_i|^2} \begin{pmatrix}\mu_1^*\cdot\mu_3&\mu_1^*\cdot\mu_4\\\mu_2^*\cdot\mu_3&\mu_2^*\cdot\mu_4\end{pmatrix} $$ where the asterisk stands for conjugate. I need to calculate the mean value of the largest eigenvalue of $\chi^\dagger\chi$ as a function of $K$ (where the dagger stands for conjugate transpose).

...I don't even know where to start from. I found this paper which I hoped could be of help, but the four entries of the matrix $\chi$, above, are not independent of each other, so I'm not sure how to use that characteristic function (eq. 2 in the paper).


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