Title: Fuglede's theorem for approximately normal matrices with uniform error estimation.

Abstract: Fuglede proved a useful result for bounded linear operators on Hilbert space. He showed that if N is normal and T commutes with N then T also commutes with N*. In numerical linear algebra one rarely has normal matrices, but rather matrices with N*N only close of NN*, and commutativity is also usually only approximate. We will discuss a softened version of Fuglede's theorem. Finally, we will explore how theorems about finite matrices can imply theorems about all bounded linear operators, opening up the possibility to do computer searches or studies on random matrices to derive results about operators on infinite-dimensional Hilbert space.