  We give an introduction to the "stable algebra of matrices" as related to
certain problems in symbolic dynamics. We consider this stable algebra
(especially, shift equivalence and strong shift equivalence) for matrices over
general rings as well as various specific rings. This algebra is of independent
interest and can be followed with little attention to the symbolic dynamics. We
include strong connectionsto algebraic K-theory and the inverse spectral
problem for nonnegative matrices. We also review key features of the
automorphism group of a shift of finite type, and the work of Kim, Roush and
Wagoner giving counterexamples to Williams' Shift Equivalence Conjecture.
