Factorization and similarity in GL(2, Z) Original Research Article

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 image 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.