  The smallest singular value and condition number play important roles in
numerical linear algebra and the analysis of algorithms. In numerical analysis
with randomness, many previous works make Gaussian assumptions, which are not
general enough to reflect the arbitrariness of the input. To overcome this
drawback, we prove the first quantitative universality for the smallest
singular value and condition number of random matrices.
  Moreover, motivated by the study of smoothed analysis that random
perturbation makes deterministic matrices well-conditioned, we consider an
analog for random matrices. For a random matrix perturbed by independent
Gaussian noise, we show that this matrix quickly becomes approximately
Gaussian. In particular, we derive an optimal smoothed analysis for random
matrices in terms of a sharp Gaussian approximation.
