I will consider random matrices in the general linear group GL(N;C) distributed according to a heat kernel measure. This may also be described as the distribution of Brownian motion in GL(N;C) starting at the identity. Numerically, the eigenvalues appear to cluster into a certain domain $\Sigma_t$ as $N$ tends to infinity. A natural candidate for the limiting eigenvalue distribution is the “Brown measure” of the limiting object, which is Biane’s ``free multiplicative Brownian motion.'' I will describe recent work with Driver and Kemp in which we compute this Brown measure. The talk will be self contained and will have lots of pictures.