  We study invariant boundary conditions for one dimensional discrete Gaussian
Markov processes, basic toy models of spatial Markov processes in statistical
mechanics. More precisely, we give a decomposition of boundary objects in a non
trivial basis from the study of a meromorphic matrix-valued function $\Phi$
(inherent to the model) and its singularities. This provides a simple algorithm
for the explicit computation of invariant measures. As an application, we give
an "eigen" version of Szeg\H{o} limit theorem for matrix valued trigonometric
polynomials.
