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A problem I'm currently considering requires me to generate (pseudo-)random Gaussian integer matrices with Gaussian integer matrix inverses. By analogy with an algorithm I know for generating random orthogonal matrices, I considered an algorithm where a random Gaussian integer matrix whose real and imaginary parts are from a uniform distribution, perform a reduction to the Hermite normal form, and then take the unimodular matrix that reduces the original random matrix to Hermite form as the result. I am aware that this method misses the matrices whose determinants are $\pm i$, but I am curious as to what probability distribution the matrices generated by this method follow.

By analogy, I know that the orthogonal matrices obtained from a QR decomposition of a matrix with normally-distributed entries follows the Haar distribution; is there a way to characterize the probability distribution of the unimodular factors of the Hermite decomposition of a uniformly-distributed random Gaussian integer matrix?

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