  We show that, under mild assumptions, the spectrum of a sum of independent
random matrices is close to that of the Gaussian random matrix whose entries
have the same mean and covariance. This nonasymptotic universality principle
yields sharp matrix concentration inequalities for general sums of independent
random matrices when combined with the Gaussian theory of Bandeira,
Boedihardjo, and Van Handel. A key feature of the resulting theory is that it
is applicable to a broad class of random matrix models that may have highly
nonhomogeneous and dependent entries, which can be far outside the mean-field
situation considered in classical random matrix theory. We illustrate the
theory in applications to random graphs, matrix concentration inequalities for
smallest singular values, sample covariance matrices, strong asymptotic
freeness, and phase transitions in spiked models.
