characterize this complexness, it is convenient to introduce [@Lobkis00; @Pnini96] the ratio of the variances of the imaginary and real parts of the eigenvector as a single statistical parameter, hereafter called the complexness parameter [@Sav06]. One should note that this parameter is characteristic of the degree of non-orthogonality of the complex modes and, therefore, is closely connected to the Petermann factor mentioned above [@Scho00]. Other studies have considered the phase rigidity, another related parameter, introduced to characterize the degree to which a general scattering wave function is complex [@Bro03; @Bul06]. Both parameters are straightforwardly deduced from one another when the phase rigidity is calculated for a single eigenvector. The main advantage of considering the complexness parameter is to reveal a physical connection between spatial and spectral statistics [@Bar05; @Sav06].

In what follows, we study the probability distribution of the complexness parameter for a generic weakly open chaotic system and its connection with the distribution of resonance widths within the RMT approach. At the first stage, we derive an expression for the complexness parameter in the weak coupling regime and establish a general relation between its average and width fluctuations. Then accounting for the essential statistical feature of spectra in chaotic 