How to list all invertible matrices over a finite field?

Question

Given an odd prime, p, and integers n and m, I would like to quickly list all invertible m x m matrices whose entries come from the finite field of size p^n. What is an efficient way to do this?

I could list all possible (p^n)^(m x m) matrices and filter for those with non-zero determinant, but this seems wasteful since it involves calculating many determinants.

By listing all lower-diagonal (L), diagonal (D), and upper diagonal matrices (U), I can list matrices with factorization LDU, but these will never have zeros on the diagonal.

Is there a simple and efficient way to list all invertible square matrices whose entries come from a finite field?

Thank you!

What libraries are you using? If you have SageMath / Maxima this *should* be easy (though I'm not too familiar with them myself). — Lambda Fairy, Jul 07 '20 at 02:15
I have written my own finite field class based on Conway Polynomials. If there is a way to do it in Sage, that would be helpful. — doogrammargood, Jul 07 '20 at 02:21
Yes, it is simple in Sage: ```for g in GL(2,GF(3)): print(g)``` — doogrammargood, Jul 07 '20 at 02:36

score 1 · Answer 1 · answered Jul 07 '20 at 02:55

I don't think filtering all matrices is a bad strategy. The fraction of nonsingular matrices is

(1 - q) (1 - q^2) ... (1 - q^m) > 1 - q - q^2 - ... - q^m > 1 - q/(1 - q),

where q = 1/(p^n). Unless p^n = 2, that's at least half of them (and I bet we could get a better bound for p^n = 2).

That being said you could do partial Gaussian elimination with each incoming row as you traverse down the search tree, which would save a factor of Theta(m) field operations.

How to list all invertible matrices over a finite field?

1 Answers1