Abstract—In this paper, novel spectral decompositions are obtained for the solutions of generalized Lyapunov
equations, which are observed in the study of controllability and observability of the state vector in deterministic
bilinear systems. The same equations are used in the stability analysis and stabilization of stochastic linear
control systems. To calculate these spectral decompositions, an iterative algorithm is proposed that uses
the residues of the resolvent of the dynamics matrix. This algorithm converges for any initial guess, for a nonsingular
and stable dynamical system. The practical significance of the obtained results is that they allow one
to characterize the contribution of individual eigen-components or their pairwise combinations to the asymptotic
dynamics of the perturbation energy in deterministic bilinear and stochastic linear systems. In particular,
the norm of the obtained eigen-components increases when frequencies of the corresponding oscillating
modes approximate each other. Thus, the proposed decompositions provide a new fundamental approach for
quantifying resonant modal interactions in a large and important class of weakly nonlinear systems.