On Schur Forms for Matrices with Simple Eigenvalues

In this paper we consider the standard Schur problem for a square matrix A, namely the similarity unitary transformation of A into upper Schur form containing the eigenvalues of A on its diagonal. Since the profound work of Issai Schur (1909), this is a fundamental issue in the theory and applications of matrices. Nevertheless, certain details concerning the Schur problem need further clarification especially in connection with the perturbation analysis of the Schur decomposition relative to perturbations in A. In particular, the concept of regular solution to the perturbed Schur form is introduced and illustrated by several examples. We also introduce the concepts of diagonally spectral matrices and of quasi-Schur condensed forms of a matrix A, and show that they may be much less sensitive to perturbations in A.