On Schur Forms for Matrices with Simple Eigenvalues

In this paper, we consider various aspects of the Schur problem for a square complex matrix A, namely the similarity unitary transformation of A into upper triangular form containing the eigenvalues of A on its diagonal. Since the profound work of I. Schur published in 1909, this has become 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 the matrix A. We consider both canonical and condensed Schur forms. Special attention is paid to matrices with simple eigenvalues. Some new concepts, such as quasi-Schur forms and diagonally spectral matrices, are also introduced and studied.