  The notions of symbolic matrix system and $\lambda$-graph system for a
subshift are generalizations of symbolic matrix and $\lambda$-graph (= finite
symbolic matrix) for a sofic shift respectively ([Doc. Math. 4(1999),
285-340]). M. Nasu introduced the notion of textile system for a pair of graph
homomorphisms to study automorphisms and endomorphisms of topological Markov
shifts ([Mem. Amer. Math. Soc. 546,114(1995)]). In this paper, we formulate
textile systems on $\lambda$-graph systems and study automorphisms on
subshifts. We will prove that for a forward automorphism $\phi$ of a subshift
$(\Lambda,\sigma)$, the automorphisms $\phi^k \sigma^n, k\ge 0, n\ge 1$ can be
explicitly realized as a subshift defined by certain symbolic matrix systems
coming from both the strong shift equivalence representing $\phi$ and the
subshift $(\Lambda,\sigma)$. As an application of this result, if an
automorphism $\phi$ of a subshift $\Lambda$ is a simple automorphism, the
dynamical system $(\Lambda, \phi \circ \sigma)$ is topologically conjugate to
the subshift $(\Lambda, \sigma).$
