  We consider the problem of learning the dynamics of autonomous linear systems
(i.e., systems that are not affected by external control inputs) from
observations of multiple trajectories of those systems, with finite sample
guarantees. Existing results on learning rate and consistency of autonomous
linear system identification rely on observations of steady state behaviors
from a single long trajectory, and are not applicable to unstable systems. In
contrast, we consider the scenario of learning system dynamics based on
multiple short trajectories, where there are no easily observed steady state
behaviors. We provide a finite sample analysis, which shows that the dynamics
can be learned at a rate $\mathcal{O}(\frac{1}{\sqrt{N}})$ for both stable and
unstable systems, where $N$ is the number of trajectories, when the initial
state of the system has zero mean (which is a common assumption in the existing
literature). We further generalize our result to the case where the initial
state has non-zero mean. We show that one can adjust the length of the
trajectories to achieve a learning rate of
$\mathcal{O}(\sqrt{\frac{\log{N}}{N})}$ for strictly stable systems and a
learning rate of $\mathcal{O}(\frac{(\log{N})^d}{\sqrt{N}})$ for marginally
stable systems, where $d$ is some constant.
