The Computation and Sensitivity of Double Eigenvalues
Ross A. Lippert, Alan S. Edelman · Advances in Computational Mathematics · 2023
In this paper, we address a problem left open by Wilkinson concerning the computation of the shortest (least squares) distance of any matrix A to a matrix  with at least one repeated eigenvalue. We have found that by casting the problem in a more geometric light, we can find an answer. This paper presents two methods which can identify  for a given A based on a geometric interpretation of the level sets of the eig function (considered as a multivalued differentiable function on the set of diagonalizable matrices). One method will be based, ultimately, on the examination of the pseudospectra of A . Additionally, relations will be given between the conditioning of the problem of  and the conditioning of the algorithms for solving that problem.