Limiting empirical spectral distribution for products of rectangular matrices

Yongcheng Qi; Hongru Zhao

doi:10.1016/j.jmaa.2021.125237

Limiting empirical spectral distribution for products of rectangular matrices

Yongcheng Qi, Hongru Zhao

Mathematics & Statistics

Research output: Contribution to journal › Article › peer-review

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Abstract

In this paper, we consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables and assume the product of the m rectangular matrices is an n by n square matrix. We study the limiting empirical spectral distributions of the product where the dimension of the product matrix goes to infinity, and m may change with the dimension of the product matrix and diverge. We give a complete description for the limiting distribution of the empirical spectral distributions for the product matrix and illustrate some examples.

Original language	English (US)
Article number	125237
Journal	Journal of Mathematical Analysis and Applications
Volume	502
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2021.125237
State	Published - Oct 15 2021

Bibliographical note

Funding Information:
The authors would like to thank an anonymous referee for his/her careful reading of the manuscript and suggestion which has improved the layout of the manuscript. The research of Yongcheng Qi was supported in part by NSF Grant DMS-1916014 .

Publisher Copyright:
© 2021 Elsevier Inc.

Keywords

Eigenvalues
Empirical spectral distribution
Non-Hermitian random matrix
Product of rectangular matrices

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N2 - In this paper, we consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables and assume the product of the m rectangular matrices is an n by n square matrix. We study the limiting empirical spectral distributions of the product where the dimension of the product matrix goes to infinity, and m may change with the dimension of the product matrix and diverge. We give a complete description for the limiting distribution of the empirical spectral distributions for the product matrix and illustrate some examples.

AB - In this paper, we consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables and assume the product of the m rectangular matrices is an n by n square matrix. We study the limiting empirical spectral distributions of the product where the dimension of the product matrix goes to infinity, and m may change with the dimension of the product matrix and diverge. We give a complete description for the limiting distribution of the empirical spectral distributions for the product matrix and illustrate some examples.

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Limiting empirical spectral distribution for products of rectangular matrices

Abstract

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