# Generating a random Eisenstein integer matrix whose inverse has Eisenstein integer entries

Thanks to a question I previously asked, I realized that a Gaussian integer matrix should have a determinant of $\pm 1$ or $\pm i$ for it to have an Gaussian integer inverse. From that, I gather that if one considers an Eisenstein integer matrix, it should have a determinant of $\pm 1$, $\pm \exp(\pm 2\pi i/3)$. For the Gaussian integer case at least, I can generate random Gaussian integer matrices through the reduction to Hermite normal form.

I now ask if there is an algorithm for generating a random Eisenstein integer matrix whose inverse has Eisenstein integer entries.

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Depends a bit on what you mean by "random". –  Gerry Myerson Jul 8 '11 at 4:38
(For Gerry; mods, please transfer this into the comments section if you can) I'm not entirely sure what probability distribution do these special Eisenstein integer matrices follow, so I didn't specify. Maybe an answerer can help me out here? –  gorilla Jul 8 '11 at 4:52
I think the point is that there is no particular distinguished probability distribution, just as there is none on the integers. –  Robert Israel Jul 8 '11 at 6:41

As Gerry pointed out, your question is a bit imprecise. However, if you simply want to generate large sets of invertible matrices with entries in the prescribed ring (Gaussian integers, or Eisensteinian integers) here's what you can do.