Abstract: The article proves that the state of a bilinear control system can be split uniquely into
generalized modes corresponding to the eigenvalues of the dynamics matrix. It is also shown that
the Gramians of controllability and observability of a bilinear system can be divided into parts
(sub-Gramians) that characterize the measure of these generalized modes and their interactions.
Furthermore, the properties of sub-Gramians were investigated in relation to modal controllability
and observability. We also propose an algorithm for computing the Gramians and sub-Gramians
based on the element-wise computation of the solution matrix. Based on the proposed algorithm,
a novel criterion for the existence of solutions to the generalized Lyapunov equation is proposed,
which allows, in some cases, to expand the domain of guaranteed existence of a solution of bilinear
equations. Examples are provided that illustrate the application and practical use of the considered
spectral decompositions.