The method of complex eigenvalues is generalized to many dimensional problems by means of the scattering matrix. It is essential to allow for the existence of a background matrix on which the resonance is superposed. The "radioactive state" (i.e., the state with complex energy for which waves in all disintegration channels are outgoing) determines the damping constants of the resonance formulas. Phase constants [Eq. (6.2)] responsible for displacements of observable resonance peaks with respect to the real part of the complex eigenvalue of the energy are also determined by the radioactive state. The background matrix is restricted by the abovementioned damping constants and phase constants to a considerable extent, leaving free in its specification an $n$-1-dimensional symmetric unitary matrix for the case of $n$ channels. The equations presented do not include the case of continuously variable energy distribution among disintegration products.