Two equalities expressing the determinant of a matrix in terms of expectations over matrix-vector products
We introduce two equations expressing the inverse determinant of a full rank matrix ๐โโ^n ร n in terms of expectations over matrix-vector products. The first relationship is |det (๐)|^-1 = ๐ผ_๐ฌโผ๐ฎ^n-1[ โ๐๐ฌโ^-n], where expectations are over vectors drawn uniformly on the surface of an n-dimensional radius one hypersphere. The second relationship is |det(๐)|^-1 = ๐ผ_๐ฑโผ q[ p(๐๐ฑ) / q(๐ฑ)], where p and q are smooth distributions, and q has full support.
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