Electronic Communications in Probability

On the spectrum of sum and product of non-hermitian random matrices

Charles Bordenave

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Abstract

In this note, we revisit the work of T. Tao and V. Vu on large non-hermitian random matrices with independent and identically distributed (i.i.d.) entries with mean zero and unit variance. We prove under weaker assumptions that the limit spectral distribution of sum and product of non-hermitian random matrices is universal. As a byproduct, we show that the generalized eigenvalues distribution of two independent matrices converges almost surely to the uniform measure on the Riemann sphere.

Article information

Source
Electron. Commun. Probab., Volume 16 (2011), paper no. 10, 104-113.

Dates
Accepted: 12 February 2011
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465261966

Digital Object Identifier
doi:10.1214/ECP.v16-1606

Mathematical Reviews number (MathSciNet)
MR2772389

Zentralblatt MATH identifier
1227.60010

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Secondary: 47A10: Spectrum, resolvent 15A18: Eigenvalues, singular values, and eigenvectors

Keywords
generalized eigenvalues non-hermitian random matrices spherical law

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Bordenave, Charles. On the spectrum of sum and product of non-hermitian random matrices. Electron. Commun. Probab. 16 (2011), paper no. 10, 104--113. doi:10.1214/ECP.v16-1606. https://projecteuclid.org/euclid.ecp/1465261966


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