  Dynamic mode decomposition (DMD) is a popular data-driven framework to
extract linear dynamics from complex high-dimensional systems. In this work, we
study the system identification properties of DMD. We first show that DMD is
invariant under linear transformations in the image of the data matrix. If, in
addition, the data is constructed from a linear time-invariant system, then we
prove that DMD can recover the original dynamics under mild conditions. If the
linear dynamics are discretized with a Runge-Kutta method, then we further
classify the error of the DMD approximation and detail that for one-stage
Runge-Kutta methods even the continuous dynamics can be recovered with DMD. A
numerical example illustrates the theoretical findings.
