We propose a data-driven framework for analyzing Controllability and Observability in both linear and nonlinear dynamical systems. Traditional notions of controllability rely on system matrices and rank conditions; our approach extends these concepts using trajectory-level data and manifold learning techniques. We employ Alternating Diffusion Maps to identify directions in state space that are jointly influenced by control inputs (controllability) or reflected in system outputs (observability). We consider case studies in low-dimensional linear systems and compare the rank of the controllability matrix to the number of effectively controlled dimensions. For nonlinear systems, local linear regression residuals in learned diffusion coordinates uncover latent controllable and observable directions. We further explore the interplay between controllability and observability by implementing output-informed Diffusion Maps and constructing data-driven balanced realizations. Our method provides a unified geometric perspective on system influence and inference, enabling model reduction and improved interpretability in complex, partially observed dynamics.
