Vol. 1, No. 4, 2019

Download this article
Download this article For screen
For printing
Recent Issues
Volume 3, Issue 2
Volume 3, Issue 1
Volume 2, Issue 4
Volume 2, Issue 3
Volume 2, Issue 2
Volume 2, Issue 1
Volume 1, Issue 4
Volume 1, Issue 3
Volume 1, Issue 2
Volume 1, Issue 1
The Journal
About the Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN (electronic): 2576-7666
ISSN (print): 2576-7658
Author Index
To Appear
Other MSP Journals
This article is available for purchase or by subscription. See below.
Horn's problem and Fourier analysis

Jacques Faraut

Vol. 1 (2019), No. 4, 585–606

Let A and B be two n × n Hermitian matrices. Assume that the eigenvalues α1,,αn of A are known, as well as the eigenvalues β1,,βn of B. What can be said about the eigenvalues of the sum C = A + B? This is Horn’s problem. We revisit this question from a probabilistic viewpoint. The set of Hermitian matrices with spectrum {α1,,αn} is an orbit Oα for the natural action of the unitary group U(n) on the space of n × n Hermitian matrices. Assume that the random Hermitian matrix X is uniformly distributed on the orbit Oα and, independently, the random Hermitian matrix Y is uniformly distributed on Oβ. We establish a formula for the joint distribution of the eigenvalues of the sum Z = X + Y . The proof involves orbital measures with their Fourier transforms, and Heckman’s measures.

PDF Access Denied

However, your active subscription may be available on Project Euclid at

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

eigenvalue, Horn's problem, Heckman's measure
Mathematical Subject Classification 2010
Primary: 15A18, 15A42
Secondary: 22E15, 42B37
Received: 16 April 2018
Revised: 5 September 2018
Accepted: 19 September 2018
Published: 14 December 2018
Jacques Faraut
Institut de Mathematiques de Jussieu
Sobronne Université