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We show that, under mild assumptions, the spectrum of a sum of independentrandom matrices is close to that of the Gaussian random matrix whose entrieshave the same mean and covariance. This nonasymptotic universality principleyields sharp matrix concentration inequalities for general sums of independentrandom matrices when combined with the Gaussian theory of Bandeira,Boedihardjo, and Van Handel. A key feature of the resulting theory is that itis applicable to a remarkably broad class of random matrix models that may havehighly nonhomogeneous and dependent entries, which can be far outside themean-field situation considered in classical random matrix theory. Weillustrate the theory in applications to random graphs, matrix concentrationinequalities for smallest singular values, sample covariance matrices, strongasymptotic freeness, and phase transitions in spiked models.

Universality and sharp matrix concentration inequalities