This paper presents unifying results for subspace identification (SID) and
dynamic mode decomposition (DMD) for autonomous dynamical systems. We observe
that SID seeks to solve an optimization problem to estimate an extended
observability matrix and a state sequence that minimizes the prediction error
for the state-space model. Moreover, we observe that DMD seeks to solve a
rank-constrained matrix regression problem that minimizes the prediction error
of an extended autoregressive model. We prove that existence conditions for
perfect (error-free) state-space and low-rank extended autoregressive models
are equivalent and that the SID and DMD optimization problems are equivalent.
We exploit these results to propose a SID-DMD algorithm that delivers a
provably optimal model and that is easy to implement. We demonstrate our
developments using a case study that aims to build dynamical models directly
from video data.