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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".2011-07-08
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    (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?2011-07-08
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    I think the point is that there is no particular distinguished probability distribution, just as there is none on the integers.2011-07-08

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