Time scale decomposition of linear implicit singularly perturbed systems

An implicit singularly perturbed system
\dot{x} = A(\epsilon)x
, where
A(0)
is singular, is considered. This note presents an algorithm for decomposing the system into a set of isolated subsystems at different time scales. The decomposed form of the system is used to construct a uniform asymptotic approximation to
\exp (A(\epsilon)t)
.