Version 1
: Received: 10 October 2024 / Approved: 10 October 2024 / Online: 10 October 2024 (10:26:14 CEST)
How to cite:
Konstantinov, M. M.; Petkov, P. H. On Schur Forms for Matrices with Simple Eigenvalues. Preprints2024, 2024100785. https://doi.org/10.20944/preprints202410.0785.v1
Konstantinov, M. M.; Petkov, P. H. On Schur Forms for Matrices with Simple Eigenvalues. Preprints 2024, 2024100785. https://doi.org/10.20944/preprints202410.0785.v1
Konstantinov, M. M.; Petkov, P. H. On Schur Forms for Matrices with Simple Eigenvalues. Preprints2024, 2024100785. https://doi.org/10.20944/preprints202410.0785.v1
APA Style
Konstantinov, M. M., & Petkov, P. H. (2024). On Schur Forms for Matrices with Simple Eigenvalues. Preprints. https://doi.org/10.20944/preprints202410.0785.v1
Chicago/Turabian Style
Konstantinov, M. M. and Petko Hristov Petkov. 2024 "On Schur Forms for Matrices with Simple Eigenvalues" Preprints. https://doi.org/10.20944/preprints202410.0785.v1
Abstract
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.
Keywords
Schur canonical form; Schur condensed form; diagonally spectral matrix; quasi-Schur form; perturbations of Schur form
Subject
Computer Science and Mathematics, Applied Mathematics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
