Generation of All Invertible Matrices

Asked Nov 23 '23 at 13:45

Active Nov 24 '23 at 06:59

Viewed 121 times

I am seeking an efficient and elegant method to construct all invertible matrices in $GL_n(F)$. While I am aware that one can randomly generate matrices and check for invertibility, I aim to systematically construct all $(q^n-1)(q^n-q^1)(q^n-q^2)...(q^n-q^{n-1})$ invertible matrices, where $F$ is finite field and $q = |F|$.

To illustrate, consider the straightforward method of constructing all subsets of a given set $S$ by numbering them from $0$ to $2^{|S|}-1$. I am hoping for a similar systematic approach, if one exists.

edited Nov 24 '23 at 06:59

asked Nov 23 '23 at 13:45

Gor Melkumyan

so one way is to generate all $F^{n\times n}$ and always check. Most are invertible, usually – Peter Franek Nov 23 '23 at 13:50
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try to do a recursive algorithm that for any $m\le n$ gives you all the full rank matrices $m\times n$ – Exodd Nov 23 '23 at 14:09
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https://math.stackexchange.com/questions/1799854/is-the-l-in-lu-factorization-unique#comment3675501_1799854
Generate them as LU-factorization, (products of L and U matrices), where the L has only 1s in the diagonal and the U has no zeroes in the diagonal.
– user3257842 Nov 23 '23 at 14:25
Please add what F is. – Martin Brandenburg Nov 23 '23 at 18:03
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F is finete field. – Gor Melkumyan Nov 24 '23 at 06:58

Generation of All Invertible Matrices

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