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).

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