Diagonalization of matrices over graded principal ideal domains

AbstractThe 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

Diagonalizing triangular matrices via orthogonal Pierce decompositions

A class of sufficient conditions is given to ensure that the sum a+b in a ring R, is equivalent to a sum x + y, which is an orthogonal Pierce decomposition. This is then used to show that a lower triangular matrix, with a regular diagonal is equivalent to its diagonal iff the matrix admits a lower triangular von Neumann inverse.Fundação para a Ciência e a Tecnologia (FCT) – Programa Operacional “Ciência, Tecnologia, Inovação” (POCTI)

Drazin invertibility for matrices over an arbitrary ring

Characterizations are given for existence of the Drazin inverse of a matrix over an arbitrary ring. Moreover, the Drazin inverse of a product PAQ for which there exist a P' and Q' such that P'PA=A=AQQ' can be characterized and computed. This generalizes recent results obtained for the group inverse of such products.http://www.sciencedirect.com/science/article/B6V0R-498V7KB-3/1/0ce4de1ffadeb127d16f63c275d6027