Estimating the Spectrum of a Matrix with Moments

Krylov subspace methods are one of the premier techniques to estimate the extreme eigenvalues of a matrix. While these methods are well-understood in the Hermitian setting, due, in part, to the existence of an orthonormal eigenbasis and the connection to orthogonal polynomials, many questions still remain in the non-Hermitian case. In this talk, I'll discuss some recent work on estimating the extreme eigenvalues of a non-Hermitian matrix, as well as ideas and barriers for more global estimation of the spectrum as a whole.