On sensitivity of eigenvalues and eigendecompositions of matrices

The main purpose of this paper is to provide solutions of two problems on sensitivity of eigenvalues and eigendecompositions of matrices. The first problem is due to Wilkinson and it concerns finding the distance from an n × n matrix with n distinct eigenvalues to the set of matrices having multiple eigenvalues. We also describe how to construct a nearest matrix having a multiple eigenvalue. The second problem concerns providing a characterization of the stability of eigendecompositions of matrices and is due to Demmel.