Hurwitz and the origins of random matrix theory in mathematics

P Diaconis; PJ Forrester

Journal article

Hurwitz and the origins of random matrix theory in mathematics

P Diaconis, PJ Forrester

Random Matrices Theory and Application | WORLD SCIENTIFIC PUBL CO PTE LTD | Published : 2017

DOI: 10.1142/S2010326317300017

Abstract

The purpose of this paper is to put forward the claim that Hurwitz's paper [Über die Erzeugung der invarianten durch integration, Nachr. Ges. Wiss. Göttingen 1897 (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 parametrization - 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 un..

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University of Melbourne Researchers

Peter Forrester Author

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Awarded by National Science Foundation

Funding Acknowledgements

The work of PD was supported by NSF Grant DMS-1208775 and the Australian Research Council Grant DP130100674, and the work of PJF was supported by the Australian Research Council Centre of Excellence ACEMS, and the Australian Research Council Grant DP140102613. Thanks are due to Sherry Wang for help in preparation of the manuscript, John Stillwell for a careful reading, Colin Mallows for his help over the years and to Arun Ram for arranging our meeting at Melbourne.