the complexness parameter are solely due to fluctuations of the resonance widths. Indeed, in the extreme case of all the widths being equal, the anti-Hermitian part of ${\mathcal{H}_{\mathrm{eff}}}$ gets proportional to the unit matrix and, as a result, the complex (biorthogonal) eigenvectors become essentially real [@Sav06]. It is, therefore, instructive to take this explicitly into account and, in view of relation (\[gmg2m\]), bring Eq. (\[Xav\]) to the form: $$\label{Xvar}
  {\left \langle X \right \rangle} = \frac{f}{2}\mathrm{var}(\gamma) \,.$$ This expression relates the average complexness parameter to the natural measure of the width fluctuations, its variance.

Strong correlations between the complexness parameter and the spectral widths are already known. The proportionality between $\sqrt{{\left \langle X \right \rangle}}$ and the average value of the fluctuating part of damping was recently found experimentally in a chaotic microwave billiard at room temperature, where this was also explained heuristically using a ray picture based on the ergodic character of the wave system [@Bar05]. Then this proportionality was established in [@Sav06] using a two-level RMT model and considering $M\gg1$ that was relevant for this experiment. Expression (\[Xvar\]) readily provides this feature, in view of $\sqrt{{\left \langle X \right \rangle}}={\left \langle \gamma \right \rangle}\sqrt{f/M}$, at any $N$ and 