Integral similarity and commutators of integral matrices
It is well known and easily proved that the trace function on a matrix algebra over a field is determined, up to scalar multiples, as that unique linear function which annihilates all commutators. In the same context, but less readily proved, is that any traceless matrix can be expressed as a commutator -- Shoda's theorem. It is shown in this paper that the result is also true if the field is replaced by the ring of integers. A related result, in the case where the matrices are over a field, is that a non-scalar traceless matrix is similar to one with zero diagonal. This theorem persists, with minor restrictions, when the field is replaced by the ring of integers. The proofs of these results involve a nice mixture of simple matrix computations, some manipulations of canonical forms, and an application of Dirichlet's theorem on the existence of primes in arithmetic progressions. The subject is not yet exhausted; indeed, the authors speculate on extensions to more general rings.
scientific article; zbMATH DE number 553981
scientific article