Block conjugacy of irreducible toral automorphisms

We establish a matrix characterization (called block conjugacy) of the notion of weak equivalence of ideals. This gives the existence of a one-to-one correspondence between equivalence classes of block conjugate toral automorphisms (of a given irreducible characteristic polynomial) and equivalence classes of the weakly equivalent associated ideals. We show that there exist nonconjugate irreducible hyperbolic toral automorphisms A and B for which and are conjugate.