Concentration of Norms and Eigenvalues of Random Matrices

In this paper we prove concentration results for norms of rectangular random matrices acting as operators between lp spaces, and eigenvalues of self-adjoint random matrices. Except for the self-adjointness condition when we consider eigenvalues, the only assumptions on the distribution of the matrix entries are independence and boundedness. Our approach is based on a powerful isoperimetric inequality for product probability spaces due to Talagrand [20]. Throughout this paper X = Xm,n will stand for an m × n random matrix with real or complex entries xjk. (Specific technical conditions on the xjk’s will be introduced as needed for each result below.) If 1 ≤ p, q ≤ ∞ and A is an m× n matrix, we denote by ‖A‖p→q the operator norm of A : lp → lq . We denote by p′ = p/(p − 1) the conjugate exponent of p. For a real random variable Y we denote by EY the expected value and by MY any median of Y . Our first main result is the following.