Two equalities expressing the determinant of a matrix in terms of expectations over matrix-vector products

Abstract

We introduce two equations expressing the inverse determinant of a full rank matrix $\mathbf{A}\in {\mathbb{R}}^{n\times n}$ in terms of expectations over matrix-vector products. The first relationship is $|\det (\mathbf{A}){|}^{-1}={\mathbb{E}}_{\mathbf{s}\sim {\mathcal{S}}^{n-1}}[\Vert \mathbf{As}{\Vert }^{-n}]$, where expectations are over vectors drawn uniformly on the surface of an $n$-dimensional radius one hypersphere. The second relationship is $|\det (\mathbf{A}){|}^{-1}={\mathbb{E}}_{\mathbf{x}\sim q}[p(\mathbf{Ax})/q(\mathbf{x})]$, where $p$ and $q$ are smooth distributions, and $q$ has full support.

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