Factorization and similarity in GL(2, Z)

Abstract

We give a solution to the question: given matrices A and A′ in GL(2, Z), when is there a third matrix C in GL(2, Z) such that AC = CA′? It is shown that any A ϵ GL(2, Z) is similar to one of a certain “standard” form, nonuniquely. Standard matrices can be factored uniquely into a product of “elementary” matrices c l l 0 as done by van der Poorten. Using the theory of continued fractions, we then show that two standard matrices are similar if and only if their factorizations are equal up to a cyclical permutation of the factors.