Re: p-adic Not at first. I'd like to see what analogies exist for GF(p) first, then GF(p^k). Only after these are understood would I go after p-adics. It's obvious that PDQ => M, where D diagonal, P,Q orthogonal can be constructed. So which square matrices M over GP(p) can be so factored? What works as a "square root" -- or more importantly, what works as a *square* to convert eigenvalues to singular values ? It is also easy to do the usual *iterations* over finite fields; the real (pun intended) question is: do these iterations do anything interesting and/or useful ? Continued fractions with square matrix "numbers" ? Are these useful for anything? Etc., etc. At 03:18 PM 2/21/2019, Fred Lunnon wrote:

Rather over p-adic fields, perhaps? WFL

On 2/21/19, Henry Baker <hbaker1@pipeline.com> wrote:

...

Are there any tutorials or cookbooks on extending some of the usual linear algebra results to finite fields?

I'm particularly interested in these algorithms on square matrices: * polar decomposition * eigenvalue decomposition * singular value decomposition * what happens when various classic *iterations* are performed over a finite field -- e.g., computing polar decomposition via iteration (does it even work?) * Are there any linear/convex programming ideas that carry over to finite fields?

I'm also interested in a cookbook of various classical Newton-style iterations, but performed over the non-commutative rings of square matrices of real & complex #'s -- in the manner of computing the polar decomposition by Newton's square root iteration. Since these square matrices are in general non-commutative, there are a lot more choices about how to extend the commutative Newton iterations to the non-commutative cases.

I suspect that relatively little work has be done on this last issue, because it's only been relatively recently that PC's have been powerful enough to do thousands/millions of iterations on entire matrix operations (as opposed to row ops or column ops).