A Note on Decomposable and Reducible Integer Matrices

Abstract

We propose necessary and sufficient conditions for an integer matrix to be decomposable in terms of its Hermite normal form. Specifically, to each integer matrix, we associate a symmetric integer matrix whose reducibility can be efficiently determined by elementary linear algebra techniques, and which completely determines the decomposability of the first one.

Keywords

• integer matrix
• hermite normal form
• decomposable matrix
• reducible matrix
• disconnected graph