Diagonalization Algorithms

I will define the problem of approximately diagonalizing a given dense matrix. I will explain two phenomena which impede the convergence of diagonalization algorithms and complicate their analysis: small eigenvalue gaps, and non-orthogonal eigenvectors (i.e., nonnormality). Finally, I will explain how random perturbations can be used to surmount these difficulties and survey what is known and remains to be discovered in this area.