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We obtain nonasymptotic bounds on the spectral norm of random matrices withindependent entries that improve significantly on earlier results. If $X$ isthe $n\times n$ symmetric matrix with $X_{ij}\sim N(0,b_{ij}^2)$, we show that\[\mathbf{E}\Vert X\Vert \lesssim\max_i\sqrt{\sum_jb_{ij}^2}+\max _{ij}\vertb_{ij}\vert \sqrt{\log n}.\] This bound is optimal in the sense that a matchinglower bound holds under mild assumptions, and the constants are sufficientlysharp that we can often capture the precise edge of the spectrum. Analogousresults are obtained for rectangular matrices and for more general sub-Gaussianor heavy-tailed distributions of the entries, and we derive tail bounds inaddition to bounds on the expected norm. The proofs are based on a combinationof the moment method and geometric functional analysis techniques. As anapplication, we show that our bounds immediately yield the correct phasetransition behavior of the spectral edge of random band matrices and of sparseWigner matrices. We also recover a result of Seginer on the norm of Rademachermatrices.

Sharp nonasymptotic bounds on the norm of random matrices with independent entries