  We define and analyze the operations of addition and intersection of linear
time-invariant systems in the behavioral setting, where systems are viewed as
sets of trajectories rather than input-output maps. The classical definition of
addition of input-output systems is addition of the outputs with the inputs
being equal. In the behavioral setting, addition of systems is defined as
addition of all variables. Intersection of linear time-invariant systems was
considered before only for the autonomous case in the context of "common
dynamics" estimation. We generalize the notion of common dynamics to open
systems (systems with inputs) as intersection of behaviors. This is done by
proposing trajectory-based definitions. The main results of the paper are 1)
characterization of the link between the complexities (number of inputs and
order) of the sum and intersection systems, 2) algorithms for computing their
kernel and image representations and 3) a duality property of the two
operations. Our approach combines polynomial and numerical linear algebra
computations.
