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
both explains the phenomenon behind classical matrix concentration inequalities
such as the matrix Bernstein inequality, and yields new sharp matrix
concentration inequalities for general sums of independent random matrices when
combined with the recent Gaussian theory of Bandeira, Boedihardjo, and Van
Handel. As an application of our main results, we prove strong asymptotic
freeness of a very general class of random matrix models with non-Gaussian,
nonhomogeneous, and dependent entries.Comment: 56 page