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Are there any known results on the Smith Normal Form for block matrices over the integers?

In particular, I am interested in matrices of size $kr \times ks$ made of square blocks of size $k$ such that each block has one $1$ and one $-1$ in each row and column and the rest are zeros. This is for computing the cohomology of a certain chain complex.

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  • $\begingroup$ What kind of results? Special properties of the Smith form in that case? Efficient computation? A "block-Smith" variant that works on the matrix ring of $k\times k$ matrices? $\endgroup$
    – Federico Poloni
    Commented Aug 22, 2014 at 10:46
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    $\begingroup$ I would like to prove that this kind of matrix have always the same type of smith normal form: an identity matrix and than only zeros. $\endgroup$
    – user53075
    Commented Aug 22, 2014 at 11:22

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On efficient sparse integer matrix Smith normal form computations (Dumas, Saunders, Villard, 2001)

We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo powers of word-size primes. Our method has proven useful in algebraic topology for the computation of the homology of some large simplicial complexes.

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