A symbolic matrix system and a $\lambda$-graph system for a subshift are generalizations of a symbolic matrix and a $\lambda$-graph (= finite symbolic matrix); for a sofic shift respectively. In [K. Matsumoto. (Presentations of subshifts and their topological conjugacy invariants. Doc. Math. 4 (1999), 285–340)], two equivalence relations between symbolic matrix systems were formulated, properly strong shift equivalence and strong shift equivalence, which are related to topological conjugacy of subshifts. In this paper, we first prove that these two equivalence relations are equivalent to each other for a large class of symbolic matrix systems. State splitting of a $\lambda$-graph system is introduced as a generalization of state splitting of a finite directed graph. State splitting gives rise to a strong shift equivalence for the corresponding symbolic matrix systems. We prove that two subshifts are topologically conjugate if and only if the canonical $\lambda$-graph system for one subshift is obtained by a finite sequence of merged state splittings and merged state amalgamations from the canonical $\lambda$-graph system for the other subshift.