# Smith Normal Form for block matrices over the integers

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.

• 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? – Federico Poloni Aug 22 '14 at 10:46
• 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. – user53075 Aug 22 '14 at 11:22