The numerical linear algebra folks have ruined the term 'factorization', because essentially all of their work involves _approximations_ to exact factors. However, if one is interested in _exact_ matrix factorization -- e.g., matrices of integers -- what is known? I wrote a simple Maxima function that removes gcd's of rows & columns, but I was wondering if I (easily) could go any further. Clearly, there are lots of questions: e.g., even for square nxn matrices, the factors might be nxm & mxn. Is this allowed? Does it help? One can always use (integer) permutation matrices to reorder rows & columns, so we won't worry about that, unless we can come up with an appropriate sorting rule. I'm sure there must be a literature on this problem, but I did a quick Google search & couldn't find the appropriate terminology.