Evaluation of divisor functions of matrices

1. Introduction. The study of divisor functions of matrices arose legitimately in the context of arithmetic of matrices, and the question of the number of (possibly weighted) inequivalent factorizations of an integer matrix was asked. However, till now only partial answers were available. Nanda [6] evaluated the case of prime matrices and Narang [7] gave an evaluation for 2×2 matrices. We obtained a recursion in the size of the matrices and the weights of the divisors [1,2] which helped us obtain a result for 3×3 matrices but no closed formula for the general case. In this paper we obtain the complete evaluation of the divisor functions by a combinatorial consideration (see Theorem 1). Because of the existence of a bijection (detailed in a forthcoming paper [3]) between the set of divisors of an r×r integer matrix and the set of subgroups of an abelian group of rank at most r, we have here a rather simple proof to obtain the number of subgroups of a finite abelian group.