  This paper proves new results on spectral and scattering theory for
matrix-valued Schr\"odinger operators on the discrete line with non-compactly
supported perturbations whose first moments are assumed to exist. In
particular, a Levinson theorem is proved, in which a relation between
scattering data and spectral properties (bound and half bound states) of the
corresponding Hamiltonians is derived. The proof is based on stationary
scattering theory with prominent use of Jost solutions at complex energies that
are controlled by Volterra-type integral equations.
