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)x(ks) made of square blocks of size k such that each block has one 1 and one -1 in each row and columns and the rest are zeros. This is for computing the cohomology of a certain chain complex.
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.