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Limiting Eigenvectors of Outliers for Spiked Information-Plus-Noise Type Matrices

  • Mireille Capitaine
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2215)

Abstract

We consider an Information-Plus-Noise type matrix where the Information matrix is a spiked matrix. When some eigenvalues of the random matrix separate from the bulk, we study how the corresponding eigenvectors project onto those of the spikes. Note that, in an Appendix, we present alternative versions of the earlier results of Bai and Silverstein (Random Matrices Theory Appl 1(1):1150004, 44, 2012) (“noeigenvalue outside the support of the deterministic equivalent measure”) and Capitaine (Indiana Univ Math J 63(6):1875–1910, 2014) (“exact separation phenomenon”) where we remove some technical assumptions that were difficult to handle.

Keywords

Random matrices Spiked information-plus-noise type matrices Eigenvalues Eigenvectors Outliers Deterministic equivalent measure Exact separation phenomenon 

Notes

Acknowledgements

The author is very grateful to Charles Bordenave and Serban Belinschi for several fruitful discussions and thanks Serban Belinschi for pointing out Lemma 4.14. The author also wants to thank an anonymous referee who provided a much simpler proof of Lemma 4.13 and encouraged the author to establish the results for non diagonal perturbations, which led to an overall improvement of the paper.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CNRS, Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouse CedexFrance

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