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Israel Journal of Mathematics

, Volume 224, Issue 1, pp 385–406 | Cite as

How many matrices can be spectrally balanced simultaneously?

  • Ronen Eldan
  • Fedor Nazarov
  • Yuval Peres
Article
  • 27 Downloads

Abstract

We prove that any ℓ positive definite d × d matrices, M1,...,M, of full rank, can be simultaneously spectrally balanced in the following sense: for any k < d such that ℓ ≤ \(\ell \leqslant \left\lfloor {\frac{{d - 1}}{{k - 1}}} \right\rfloor \), there exists a matrix A satisfying \(\frac{{{\lambda _1}\left( {{A^T}{M_i}A} \right)}}{{Tr\left( {{A^T}{M_i}A} \right)}} < \frac{1}{k}\) 1/k for all i, where λ1(M) denotes the largest eigenvalue of a matrix M. This answers a question posed by Peres, Popov and Sousi ([PPS13]) and completes the picture described in that paper regarding sufficient conditions for transience of self-interacting random walks. Furthermore, in some cases we give quantitative bounds on the transience of such walks.

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References

  1. [HJ13]
    R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013.zbMATHGoogle Scholar
  2. [Kat76]
    T. Kato, Perturbation Theory for Linear Operators, Grundlehren der Mathematischen Wissenschaften, Vol. 132 Springer-Verlag, Berlin–New York, 1976.Google Scholar
  3. [PPS13]
    Y. Peres, S. Popov and P. Sousi, On recurrence and transience of self-interacting random walks, Bulletin of the Brazilian Mathematical Society 44 (2013), 841–867.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Department of MathematicsThe Weizmann Institute of ScienceRehovotIsrael
  2. 2.Department of MathematicsKent State UniversityKentUSA
  3. 3.Microsoft ResearchOne Microsoft WayRedmondUSA

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