Singular Value and Eigenvalue Distribution of a Matrix-Sequence

For fixed l≥0 and m≥1, let Xn0, Xn1,..., Xnl be independent random n × n matrices with independent entries, let Fn0 := Xn0, Xn1-1,..., Xnl-1, and let Fn1,..., Fnm be independent random matrices of the same form as Fn0 . We show that as n → ∞, the matrices Fn0 and m−l+1/2Fn1 +...+ Fnm have the same limiting eigenvalue distribution. To obtain our results, we apply the general framework recently introduced in Götze, Kösters, and Tikhomirov 2015 to sums of products of independent random matrices and their inverses.We establish the universality of the limiting singular value and eigenvalue distributions, and we provide a closer description of the limiting distributions in terms of free probability theory.

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Smallest singular value and limit eigenvalue distribution of a class of non-Hermitian random matrices with statistical application

Journal of Multivariate Analysis ◽

10.1016/j.jmva.2020.104623 ◽

2020 ◽

Vol 178 ◽

pp. 104623

Author(s):

Arup Bose ◽

Walid Hachem

Keyword(s):

Random Matrices ◽

Singular Value ◽

Eigenvalue Distribution ◽

Statistical Application ◽

Smallest Singular Value

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Multilevel symmetrized Toeplitz structures and spectral distribution results for the related matrix sequences

Electronic Journal of Linear Algebra ◽

10.13001/ela.2021.5775 ◽

2021 ◽

Vol 37 ◽

pp. 370-386

Author(s):

Paola Ferrari ◽

Isabella Furci ◽

Stefano Serra-Capizzano

Keyword(s):

Toeplitz Matrix ◽

Integrable Function ◽

Toeplitz Matrices ◽

Singular Value ◽

Value Distribution ◽

Identity Matrix ◽

Matrix Size ◽

Lebesgue Integrable Function ◽

Matrix Sequence ◽

The Matrix

In recent years, motivated by computational purposes, the singular value and spectral features of the symmetrization of Toeplitz matrices generated by a Lebesgue integrable function have been studied. Indeed, under the assumptions that $f$ belongs to $L^1([-\pi,\pi])$ and it has real Fourier coefficients, the spectral and singular value distribution of the matrix-sequence $\{Y_nT_n[f]\}_n$ has been identified, where $n$ is the matrix size, $Y_n$ is the anti-identity matrix, and $T_n[f]$ is the Toeplitz matrix generated by $f$. In this note, the authors consider the multilevel Toeplitz matrix $T_{\bf n}[f]$ generated by $f\in L^1([-\pi,\pi]^k)$, $\bf n$ being a multi-index identifying the matrix-size, and they prove spectral and singular value distribution results for the matrix-sequence $\{Y_{\bf n}T_{\bf n}[f]\}_{\bf n}$ with $Y_{\bf n}$ being the corresponding tensorization of the anti-identity matrix.

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