losses [@Fyo05].

By now, complex eigenvalues of such non-Hermitian random matrices have been studied quite systematically [@Fyo97; @Fyo99; @Fyo03]. However, the statistical properties of the corresponding (left and right) eigenvectors are less understood. Quite a substantial progress in this direction has been achieved by Schomerus *et al.* [@Scho00], who studied mainly the systems with broken time-reversal symmetry. Other analytical results for a few physically interesting particular cases have also been reported in the literature recently [@Meh01; @Fyo02; @Bro03; @Sav06]. Components of eigenvectors appear as residues of the $S$ matrix at resonance positions and the understanding of their properties is thus important for many applications. For example, nonorthogonality of resonance eigenstates yields the enhancement (the so-called Petermann factor) of the line width of a lasing mode in open resonators [@Scho00] and influences branching ratios of nuclear cross-sections [@Sok97ii; @Oko03]. It features also in the particle escape from the scattering region [@Sav97] as well as in dissipative quantum chaotic maps [@Kea08].

This paper focuses on spectral and eigenvector statistics of such non-Hermitian random matrices describing effective Hamiltonians of open chaotic wave systems whose closed limit displays time-reversal symmetry (TRS). In this case, the quasimodes correspond to the complex-valued eigenvectors of ${\mathcal{H}_{\mathrm{eff}}}$. To 