We revisit the canonical continuous-time and discrete-time matrix algebraic
and matrix differential equations that play a central role in Lyapunov based
stability arguments. The goal is to generalize and extend these types of
equations and subsequent analysis to dynamical systems on domains other than
$\R$ or $\Z$, e.g. nonuniform discrete domains or domains consisting of a
mixture of discrete and continuous components. We compare and contrast the
standard theory with the theory in this general case.