A. Hurwitz and the origins of random matrix theory in mathematics

TL;DR

Hurwitz's 1897 paper laid the groundwork for random matrix theory by introducing invariant measures for matrix groups and developing a calculus for explicit computation, influencing subsequent foundational research.

Contribution

The paper highlights Hurwitz's pioneering role in establishing invariant measures and explicit integration techniques that underpin modern random matrix theory.

Findings

Invariant measures for SO(N) and U(N) derived by Hurwitz

Explicit computation of group integrals using Euler angles

Probabilistic interpretation of matrix Euler angles as independent beta distributions

Abstract

The purpose of this article is to put forward the claim that Hurwitz's paper "Uber die Erzeugung der Invarianten durch Integration." [Gott. Nachrichten (1897), 71-90] should be regarded as the origin of random matrix theory in mathematics. Here Hurwitz introduced and developed the notion of an invariant measure for the matrix groups $SO(N)$ and $U(N)$. He also specified a calculus from which the explicit form of these measures could be computed in terms of an appropriate parametrisation - Hurwitz chose to use Euler angles. This enabled him to define and compute invariant group integrals over $SO(N)$ and $U(N)$. His main result can be interpreted probabilistically: the Euler angles of a uniformly distributed matrix are independent with beta distributions (and conversely). We use this interpretation to give some new probability results. How Hurwitz's ideas and methods show themselves in the subsequent work of Weyl, Dyson and others on foundational studies in random matrix theory is detailed.