  A central tool in the study of nonhomogeneous random matrices, the
noncommutative Khintchine inequality, yields a nonasymptotic bound on the
spectral norm of general Gaussian random matrices $X=\sum_i g_i A_i$ where
$g_i$ are independent standard Gaussian variables and $A_i$ are matrix
coefficients. This bound exhibits a logarithmic dependence on dimension that is
sharp when the matrices $A_i$ commute, but often proves to be suboptimal in the
presence of noncommutativity. In this paper, we develop nonasymptotic bounds on
the spectrum of arbitrary Gaussian random matrices that can capture
noncommutativity. These bounds quantify the degree to which the spectrum of $X$
is captured by that of a noncommutative model $X_{\rm free}$ that arises from
free probability theory. This "intrinsic freeness" phenomenon provides a
powerful tool for the study of various questions that are outside the reach of
classical methods of random matrix theory. Our nonasymptotic bounds are easily
applicable in concrete situations, and yield sharp results in examples where
the noncommutative Khintchine inequality is suboptimal. When combined with a
linearization argument, our bounds imply strong asymptotic freeness for a
remarkably general class of Gaussian random matrix models that may be very
sparse, have dependent entries, and lack any special symmetries. When combined
with a universality principle, our bounds extend beyond the Gaussian setting to
general sums of independent random matrices.
