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.
Highlights
Apart from practical computational considerations, we emphasize that, given an integer matrix A, we propose a new approach by associating A with a simple graph whose connectivity determines its decomposition
We finalize this note by giving an algorithm for the computation of the decomposition of an m × n integer matrix into the direct sum matrices in Hermite normal form
Using the Hermite normal form as the main tool, we have obtained a theoretical criterion to determine whether a given integer matrix decomposes into a direct sum of lower order integer matrices
Summary
We can adapt the combinatorial and linear algebra machinery to determine if A is decomposable: note that, for a symmetric real matrix, it is possible to decide if it can be decomposed into a direct sum of smaller symmetric real matrices by analyzing the connectivity of a certain associated graph, which is closely related to the spectral properties of the graph. All this allows us to propose an algorithm (Algorithm 1) for the computation of the decomposition of the matrix A, if possible.
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