Appendix

$g_i$s are complex functions. One can therefore write $S$ in terms of $g_i^I$ and $g_i^R$, the real and imaginary parts of $g_i$ and minimize with respect to $g_i^I$ and $g_i^R$ separately. It is shown here that the complex arithmetic achieves exactly this and the results are same as that given by complex calculus. The superscripts $I$ and $R$ in the following are used to represent the real and imaginary parts of complex quantities.

Expanding Eq. 5, ignoring $w_{ij}$s and writing it in terms of real and imaginary parts we get

$$\begin{split}\sum\limits_{{i,j} \atop {i \ne j}}\left\vert X_... ...=\sum\limits_{{i,j} \atop {i \ne j}}& S_0 S_0^\star \end{split}$$ (14)

where

$$S_0=\left[X_{ij}^R- g_i^Rg_j^R- g_i^Ig_j^I\right] +\iota \left[X_{ij}^I+ g_i^Rg_j^I- g_i^Ig_j^R\right]$$ (15)

Taking partial derivative of $S$ with respect to $g_i^R$ and reintroducing $w_{ij}$, we get

$$\begin{split}{\partial S \over \partial g_i^R}=&\sum\limits_{... ...g_j^I- g_i^R{g_j^I}^2 - g_i^R{g_j^R}^2\right]w_{ij} \end{split}$$ (16)

Therefore,

$${\partial S \over \partial g_i^R}= -2\sum\limits_{j \atop {j \ne i}}\left[Re(X_{ij}g_j) - \left\vert g_j\right\vert^2 g_i^R\right]w_{ij}$$ (17)

Equating $\partial S \over \partial g_i^R$ to zero, we get

$$g_i^R= {\sum\limits_{j \atop {j \ne i}}Re(X_{ij}g_j w_{ij}) \over {\sum\limits_{j \atop {j \ne i}}\left\vert g_j \right\vert^2 w_{ij}}}$$ (18)

Similarly

$${\partial S \over \partial g_i^I}=-2\sum\limits_{j \atop {j \ne i}}\left[Im(X_{ij}g_j) - \left\vert g_j\right\vert^2 g_i^I\right] w_{ij}$$ (19)

Therefore the equivalent imaginary part of Eq. 18 is

$$g_i^I= {\sum\limits_{j \atop {j \ne i}}Im(X_{ij}g_j w_{ij}) \over {\sum\limits_{j \atop {j \ne i}}\left\vert g_j \right\vert^2 w_{ij}}}$$ (20)

writing $g_i=g_i^R+ \iota g_i^I$ and substituting for $g_i^R$ and $g_i^I$ from Eq. 18 and 20 respectively, we get

$$g_i = {\sum\limits_{j \atop {j \ne i}}X_{ij}g_j w_{ij} \over {\sum\limits_{j \atop {j \ne i}}\left\vert g_j \right\vert^2 w_{ij}}}$$ (21)

This is same as Eq. 8, which was arrived at by evaluating a complex derivative of Eq. 5 as $\partial S/\partial g_i^\star$, treating $g_i$ and $g_I^\star$ as independent variables. Evaluating ${\partial S \over \partial g_i}=0$ would give the complex conjugate of Eq. 21. Hence, $\partial S/\partial g_i$ gives no independent information not present in $\partial S/\partial g_i^\star$.