eigenvectors, the diagonalization of ${\mathcal{H}_{\mathrm{eff}}}$ then reads: $$\label{eig}
  {\mathcal{H}_{\mathrm{eff}}}{|  \psi_n \rangle} = (E_n - \frac{i}{2} \Gamma_n){|  \psi_n \rangle}$$ where $E_n$ and $\Gamma_n$ are, respectively, the energy and the width of the $n$-th resonance. Due to TRS present, ${\mathcal{H}_{\mathrm{eff}}}$ is a complex symmetric matrix; hence, the left and right eigenvectors are related by the transpose, ${ \langle \tilde{\psi}_n |}=({| \psi_n \rangle})^T$ [@Sok89].

The coupling to continuum, as described by the imaginary part of ${\mathcal{H}_{\mathrm{eff}}}$, turns real eigenfunctions of the closed system into complex quasimodes of its open counterpart. In order to measure their complexness, we define the complexness parameter $q_n^2$ as follows: $$\label{qdef}
  q_n^2 = \frac{\sum_i({\textrm{Im}}\; \psi_n^i)^2}{\sum_i({\textrm{Re}}\; \psi_n^i)^2}$$ where $\psi_n^i$ is the $i$-th component of the eigenvector (we note that the complexness parameter can be equivalently defined by means of the left eigenvectors). It is worth noting here that in contrast to the related Petermann factor [@Scho00], which is defined for a fixed value of the given resonance width, no additional constraints are imposed on (\[qdef\]). In chaotic systems, $q_n^2$ reveals strong mode-to-mode fluctuations, which we describe through its probability distribution function to be derived below.

Statistical assumptions
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Within the RMT approach, the universal statistical properties of closed chaotic systems 