  We consider space discretizations of the matrix Zakharov-Shabat AKNS scheme,
in particular the discrete matrix non-linear Scrhr\"odinger (DNLS) model, and
the matrix generalization of the Ablowitz-Ladik (AL) model, which is the more
widely acknowledged discretization. We focus on the derivation of solutions via
local Darboux transforms for both discretizations, and we derive novel
solutions via generic solutions of the associated discrete linear equations.
The continuum analogue is also discussed, and as an example we identify
solutions of the matrix NLS equation in terms of the heat kernel. In this frame
we also derive a discretization of the Burgers equation via the analogue of the
Cole-Hopf transform. Using the basic Darboux transforms for each scheme we
identify both matrix DNLS-like and AL hierarchies, i.e. we extract the
associated Lax pairs, via the dressing process. We also discuss the global
Darboux transform, which is the discrete analogue of the integral transform,
through the discrete Gelfand-Levitan-Marchenko (GLM) equation. The derivation
of the discrete matrix GLM equation and associated solutions are also presented
together with explicit linearizations. Particular emphasis is given in the
discretization schemes, i.e. forward/backward in the discrete matrix DNLS
scheme versus symmetric in the discrete matrix AL model.
