Eigenvectors of a matrix under random perturbation

Benaych-Georges, Florent; Enriquez, Nathanaël; Michaïl, Alkéos

doi:10.1142/s2010326321500234

Cited by 5 publications

(4 citation statements)

References 19 publications

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“…Although the argument is standard and can be found for the empirical spectral measure of a Wigner matrix in the survey of Benaych-Georges and Knowles [7], we choose to provide the details as it has not been done for the spectral measures. A similar argument is also present in the work of Benaych-Georges, Enriquez and Michaïl [6].…”

Section: Convergence Of the Averaged Square Projectionssupporting

confidence: 80%

“…These are the content of Theorems 2 and 5. Our proof is inspired by the work of Benaych-Georges, Enriquez and Michaïl [6] and uses local laws estimates recently obtained by Knowles and Yin in [15]. When θ belongs to the support of the asymptotic spectrum of A n (resp.…”

Section: Introductionmentioning

confidence: 99%

“…Let us finally say a few words about previous use of spectral measures in the literature. Benaych-Georges, Enriquez and Michaïl, in [6], obtained informations on the eigenvectors of a diagonal deterministic matrix perturbed by a random symmetric matrix. In a series of works (the most recent being [14]), Gamboa, Nagel and Rouault studied spectral measures of some classical ensembles of random matrix theory and their connections with sum rules.…”

Section: Introductionmentioning

confidence: 99%

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Spectral Measures of Spiked Random Matrices

Noiry

2020

J Theor Probab

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We study two spiked models of random matrices under general frameworks corresponding respectively to additive deformation of random symmetric matrices and multiplicative perturbation of random covariance matrices. In both cases, the limiting spectral measure in the direction of an eigenvector of the perturbation leads to old and new results on the coordinates of eigenvectors.MSC 2010 Classification: 60B20.

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Section: Convergence Of the Averaged Square Projectionssupporting

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Spectral Measures of Spiked Random Matrices

Noiry

2020

J Theor Probab

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“…Assuming that W is a GOE matrix (i.e. with a Gaussian law), the law of M (t) is governed by Dyson Brownian motion at time t starting from the diagonal matrix D. We refer for instance to [23,25,64,75,76]. Although not stated explicitly in these works, the analysis of Dyson Brownian motion, as for instance in [25], implies that, with high probability, for any x ∈ W, the eigenvector associated with the eigenvalue λ(t, x) of the W × W matrix M (t) remains localized for t ≪ |W| ∆(x) We remark that the condition (F.2) is strictly weaker than Mott's criterion, which reads t ≪ ∆(x) 2 .…”

Section: A Graph Properties -Proofs Of Propositions 31 and 32mentioning

confidence: 99%

The completely delocalized region of the Erdős-Rényi graph

Alt

Ducatez

Knowles

2022

Electron. Commun. Probab.

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We analyse the eigenvectors of the adjacency matrix of the Erdős-Rényi graph on N vertices with edge probability d N . We determine the full region of delocalization by determining the critical values of d log N down to which delocalization persists: for d log N > 1 log 4−1 all eigenvectors are completely delocalized, and for d log N > 1 all eigenvectors with eigenvalues away from the spectral edges are completely delocalized. Below these critical values, it is known [1, 3] that localized eigenvectors exist in the corresponding spectral regions.

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Probabilistic perturbation bounds of matrix decompositions

Petkov

2024

Numerical Linear Algebra App

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In this article, we determine probabilistic approximations of the entries of random perturbation matrices implementing the Markoff inequality. These approximations are used to derive with prescribed probability asymptotic componentwise perturbation bounds of some orthogonal and unitary matrix decompositions. We show that the probabilistic asymptotic bounds are significantly less conservative than the corresponding deterministic perturbation bounds. As case studies, we consider the determining of probabilistic perturbation bounds of the QR decomposition, the singular value decomposition and the Schur decomposition of a matrix using an unified method for asymptotic componentwise perturbation analysis of these decompositions. It is demonstrated that the probability bounds of the orthogonal transformations, singular values and eigenvalues are much tighter than the corresponding deterministic asymptotic bounds. The probabilistic bounds derived are appropriate for perturbation analysis of large‐order matrix problems.

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Eigenvectors of a matrix under random perturbation

Cited by 5 publications

References 19 publications

Spectral Measures of Spiked Random Matrices

Spectral Measures of Spiked Random Matrices

The completely delocalized region of the Erdős-Rényi graph

Probabilistic perturbation bounds of matrix decompositions

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