Spectral Decomposition of Discrete-Time Controllability Gramian and Its Inverse via System Eigenvalues

Abstract

This paper develops a closed-form spectral decomposition framework for the Gramian matrices of discrete-time linear dynamical systems. The main results provide explicit decompositions of the discrete-time controllability Gramian and its inverse in terms of the eigenvalues of the dynamics matrix, yielding a mode-resolved representation of these matrices. In contrast to the more common use of aggregate Gramian characteristics, such as eigenvalues, singular values, determinants, and trace-based metrics, the proposed approach describes the internal structure of the Gramian itself through contributions associated with individual modes and their pairwise combinations. The framework is extended further to the solution of the discrete-time Lyapunov difference equation, placing the obtained formulas in a broader context relevant to the analysis and computation of time-varying and nonlinear systems. In addition, the decomposition is generalized to systems whose dynamics matrix has multiple eigenvalues, enabling a closed-form estimation of the effects of resonant interactions between eigenmodes. The proposed results provide a structural tool for the analysis of controllability, observability and stability in discrete-time systems and complement existing Gramian-based methods used in model reduction, estimation, actuator and sensor selection, and energy-aware control. Beyond their theoretical interest, the derived decompositions may support the development of improved computational procedures and more informative performance criteria for a range of discrete-time control problems.