  Linear time-invariant systems are very popular models in system theory and
applications. A fundamental problem in system identification that remains
rather unaddressed in extant literature is to leverage commonalities amongst
related linear systems to estimate their transition matrices more accurately.
To address this problem, the current paper investigates methods for jointly
estimating the transition matrices of multiple systems. It is assumed that the
transition matrices are unknown linear functions of some unknown shared basis
matrices. We establish finite-time estimation error rates that fully reflect
the roles of trajectory lengths, dimension, and number of systems under
consideration. The presented results are fairly general and show the
significant gains that can be achieved by pooling data across systems in
comparison to learning each system individually. Further, they are shown to be
robust against model misspecifications. To obtain the results, we develop novel
techniques that are of interest for addressing similar joint-learning problems.
They include tightly bounding estimation errors in terms of the
eigen-structures of transition matrices, establishing sharp high probability
bounds for singular values of dependent random matrices, and capturing effects
of misspecified transition matrices as the systems evolve over time.
