  This paper addresses two fundamental problems in the context of jump linear
systems (JLS). The first problem is concerned with characterizing the minimal
state space dimension solely from input-output pairs and without any knowledge
of the number of mode switches. The second problem is concerned with
characterizing the number of discrete modes of the JLS. For the first problem,
we develop a linear system theory based approach and construct an appropriate
Hankel-like matrix. The rank of this matrix gives us the state space dimension.
For the second problem we show that minimal number of modes corresponds to the
minimal rank of a positive semi-definite matrix obtained via a non--convex
formulation.
