Foundations of Computational Mathematics

, Volume 12, Issue 4, pp 389–434

User-Friendly Tail Bounds for Sums of Random Matrices

Open AccessArticle

DOI: 10.1007/s10208-011-9099-z

Cite this article as:
Tropp, J.A. Found Comput Math (2012) 12: 389. doi:10.1007/s10208-011-9099-z


This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales.

In other words, this paper provides noncommutative generalizations of the classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid. The matrix inequalities promise the same diversity of application, ease of use, and strength of conclusion that have made the scalar inequalities so valuable.


Discrete-time martingaleLarge deviationProbability inequalityRandom matrixSum of independent random variables

Mathematics Subject Classification (2000)

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© The Author(s) 2011

Authors and Affiliations

  1. 1.Computing & Mathematical Sciences, MC 305-16California Institute of TechnologyPasadenaUSA