Suppose one wants to calculate the eigenvalues of a large, non-normal matrix. For example, consider the matrix which is 0 in most places except above the diagonal, where it is 1. The eigenvalues of this matrix are all 0. Similarly, if one conjugates this matrix, in exact arithmetic one would get all eigenvalues equal to 0. However, when one makes floating point errors, the eigenvalues of this matrix are dramatically different. One can model these errors as performing a small, random perturbation to the matrix. And, far from being random, the eigenvalues of this perturbed matrix nearly exactly equidistribute on the unit circle. This talk will give a probabilistic explanation of why this happens and discuss the general question: how does one predict the eigenvalues of a large, non-normal, randomly perturbed matrix?