  In this paper, we study the nonlinear inverse problem of estimating the
spectrum of a system matrix, that drives a finite-dimensional affine dynamical
system, from partial observations of a single trajectory data. In the noiseless
case, we prove an annihilating polynomial of the system matrix, whose roots are
a subset of the spectrum, can be uniquely determined from data. We then study
which eigenvalues of the system matrix can be recovered and derive various
sufficient and necessary conditions to characterize the relationship between
the recoverability of each eigenvalue and the observation locations. We propose
various reconstruction algorithms, with theoretical guarantees, generalizing
the classical Prony method, ESPIRIT, and matrix pencil method. We test the
algorithms over a variety of examples with applications to graph signal
processing, disease modeling and a real-human motion dataset. The numerical
results validate our theoretical results and demonstrate the effectiveness of
the proposed algorithms, even when the data did not follow an exact linear
dynamical system.
