Electronic Communications in Probability

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

Charles Bordenave

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

generalized eigenvalues non-hermitian random matrices spherical law

This work is licensed under aCreative Commons Attribution 3.0 License.


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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