Diagonalization of matrices over graded principal ideal domains

Abstract

The main result of this paper is the analogue of the classical diagonal reduction of matrices over PIDs, for graded principal ideal domains. A method for diagonalizing graded matrices over a graded principal ideal domain is obtained. In Section 2 we emphasis on some applications. A procedure is given to decide whether or not a matrix defined over an ordinary Dedekind domain (i.e. nongraded), with cyclic class group, is diagonalizable. In case the answer is positive the diagonal form can be calculated. This can be done by taking a suitable graded PID which has the Dedekind domain as its part of degree zero. It turns out that, even in the case where diagonalization of a matrix over the part of degree zero is not possible, the diagonal representation over the graded ring contains useful information. The main reason for this is that the graded ring hasn't essentially more units than its part of degree zero. We illustrate this by considering the problem of von Neumann regularity of a matrix over a Gr-PID and to matrices over Dedekind domains with cyclic class group. These problems were the original motivation for studying diagonalization over graded rings.