This paper derives exponential concentration inequalities and polynomial moment inequalities for the spectral norm of a random matrix. The analysis requires a matrix extension of the scalar concentration theory developed by Sourav Chatterjee using Stein's method of exchangeable pairs. When applied to a sum of independent random matrices, this approach yields matrix generalizations of the classical inequalities due to Hoeffding, Bernstein, Khintchine and Rosenthal. The same technique delivers bounds for sums of dependent random matrices and more general matrix-valued functions of dependent random variables.Comment: Published in at http://dx.doi.org/10.1214/13-AOP892 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

This paper derives exponential concentration inequalities and polynomial moment inequalities for the spectral norm of a random matrix. The analysis requires a matrix extension of the scalar concentration theory developed by Sourav Chatterjee using Stein's method of exchangeable pairs. When applied to a sum of independent random matrices, this approach yields matrix generalizations of the classical inequalities due to Hoeffding, Bernstein, Khintchine, and Rosenthal.The same technique delivers bounds for sums of dependent random matrices and more general matrix-valued functions of dependent random variables.This paper is based on two independent manuscripts from mid-2011 that both applied the method of exchangeable pairs to establish matrix concentration inequalities. One manuscript is by Mackey and Jordan; the other is by Chen, Farrell, and Tropp. The authors have combined this research into a single unified presentation, with equal contributions from both groups.

We determine the limiting empirical singular value distribution for discrete Fourier transform (DFT) matrices when a random set of columns and rows is removed.

Abstract. A fundamental result of free probability theory due to Voiculescu and subsequently refined by many authors states that conjugation by independent Haar-distributed random unitary matrices delivers asymptotic freeness. In this paper we exhibit many other systems of random unitary matrices that, when used for conjugation, lead to freeness. We do so by first proving a general result asserting "asymptotic liberation" under quite mild conditions, and then we explain how to specialize these general results in a striking way by exploiting Hadamard matrices. In particular, we recover and generalize results of the second-named author concerning the limiting distribution of singular values of a randomly chosen submatrix of a discrete Fourier transform matrix.

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