The authors establish level dynamics for non-Hermitian \(N\times N\) matrices that may represent generators of dissipative quantum dynamics or, in another context describe properties of a scattering system. As a result the level dynamics take the form of the classical Hamiltonian flow of some fictitious \(N\)-particle system on two-dimensional plane. The authors describe equilibrium statistical mechanics of the above system and show that it leads to the well known matrix ensembles of random-matrix theory, for example, Ginibre’s ensemble.