That's related to several questions that have interested me for decades. Here's a simple version: choose your favorite matrix ring R, e.g. real matrices with entries that are 0 or 1. What is a(n) := number of nxn matrices in R that have a square root in R? About the question of whether non-square factors help, of course they sometimes do. One situation where they do is when you are given an nxn Gram matrix A for a lattice, with integer entries, and you want to know if there is an mxn matrix B with integer entries such that B transpose(B) = A. There is a lot about this in a paper I wrote with John Conway: Low-Dimensional Lattices V: Integral Coordinates for Integral Lattices, J. H. Conway and N. J. A. Sloane, Proc. Royal Soc. London, Series A, 426 (1989), pp. 211-232. (available as #151 on my pub. list on my home page, see below) If n = 1, the answer is you can do it with m = 4: this is exactly Jacobi's 4-square theorem! Neil On Tue, May 7, 2013 at 12:38 PM, Henry Baker <hbaker1@pipeline.com> wrote:
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.
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-- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com