Spectral Radii of Products of Random Rectangular Matrices

Yongcheng Qi; Mengzi Xie

doi:10.1007/s10959-019-00942-9

Spectral Radii of Products of Random Rectangular Matrices

Yongcheng Qi, Mengzi Xie

Mathematics & Statistics

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

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Abstract

We consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables. Assume the product of the m rectangular matrices is an n-by-n square matrix. The maximum absolute value of the n eigenvalues of the product matrix is called spectral radius. In this paper, we study the limiting spectral radii of the product when m changes with n and can even diverge. We give a complete description for the limiting distribution of the spectral radius. Our results reduce to those in Jiang and Qi (J Theor Probab 30(1):326–364, 2017) when the rectangular matrices are square.

Original language	English (US)
Pages (from-to)	2185-2212
Number of pages	28
Journal	Journal of Theoretical Probability
Volume	33
Issue number	4
DOIs	https://doi.org/10.1007/s10959-019-00942-9
State	Published - Dec 1 2020

Bibliographical note

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

Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

Eigenvalue
Non-Hermitian random matrix
Random rectangular matrix
Spectral radius

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abstract = "We consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables. Assume the product of the m rectangular matrices is an n-by-n square matrix. The maximum absolute value of the n eigenvalues of the product matrix is called spectral radius. In this paper, we study the limiting spectral radii of the product when m changes with n and can even diverge. We give a complete description for the limiting distribution of the spectral radius. Our results reduce to those in Jiang and Qi (J Theor Probab 30(1):326–364, 2017) when the rectangular matrices are square.",

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N2 - We consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables. Assume the product of the m rectangular matrices is an n-by-n square matrix. The maximum absolute value of the n eigenvalues of the product matrix is called spectral radius. In this paper, we study the limiting spectral radii of the product when m changes with n and can even diverge. We give a complete description for the limiting distribution of the spectral radius. Our results reduce to those in Jiang and Qi (J Theor Probab 30(1):326–364, 2017) when the rectangular matrices are square.

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Spectral Radii of Products of Random Rectangular Matrices

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