  In this article, we investigate systems of generalized Schr\"odinger
operators and their fundamental matrices. More specifically, we establish the
existence of such fundamental matrices and then prove sharp upper and lower
exponential decay estimates for them. The Schr\"odinger operators that we
consider have leading coefficients that are bounded and uniformly elliptic,
while the zeroth-order terms are assumed to be nondegenerate and belong to a
reverse H\"older class of matrices. In particular, our operators need not be
self-adjoint. The exponential bounds are governed by the so-called upper and
lower Agmon distances associated to the reverse H\"older matrix that serves as
the potential function. Furthermore, we thoroughly discuss the relationship
between this new reverse H\"{o}lder class of matrices, the more classical
matrix $\mathcal{A}_{p,\infty}$ class, and the matrix $\mathcal{A}_{\infty}$
class introduced in [Dall15].
