Pure Math Colloquium

Florent Benaych-Georges, University Paris 5

“The Single Ring Theorem”

The singular values of a complex matrix are the eigenvalues of the positive part of its polar decomposition. While the eigenvalues of the matrix are usually related to its algebraic properties, the singular values are related to its analytic properties (the largest one is its operator norm, the smallest one, when positive, is the inverse of the norm of the inverse of the matrix, etc...).

Though there are relationships between eigenvalues and singular values, neither are the singular values determined by the eigenvalues, nor are the eigenvalues determined by the singular values. How- ever, if one considers large isotropic random matrices (isotropic matrices are random non-Hermitian matrices which are invariant, in law, under the left and right actions of the unitary group), then the Single Ring Theorem, proved by Guionnet, Krishnapur and Zeitouni in 2012 (but conjectured for long by physicists), gives an asymptotic relation between the singular values and the eigenvalues, partly based on free probability theory. In this talk, we will state this result, explain some ideas used in its proof, and then present some of its recent extensions, especially on the boundary of the spectrum and at local level in the bulk of the spectrum.

MC 5501