Several Anzahl theorems of matrices over Galois rings and their applications

摘要

Let R denote the Galois ring of characteristic p(s) and cardinality p(sh). In this paper, we determine the Smith normal forms of matrices over R, compute the number of the orbits of m x n matrices under the group GL(m) (R) x GL(n) (R) and the length of each orbit. Moreover, we discuss their applications to association schemes, systems of linear equations, generalized invertible matrices, idempotent matrices and involutory matrices. (C) 2014 Elsevier Inc. All rights reserved.