On Schur Forms for Matrices with Simple Eigenvalues
In this paper, we consider various aspects of the Schur problem for a square complex matrix <i>A</i>, namely the similarity unitary transformation of <i>A</i> into upper triangular form containing the eigenvalues of <i>A</i> on its diagonal. Since the profound wor...
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2024-11-01
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| author | Mihail Mihaylov Konstantinov Petko Hristov Petkov |
| author_facet | Mihail Mihaylov Konstantinov Petko Hristov Petkov |
| author_sort | Mihail Mihaylov Konstantinov |
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| description | In this paper, we consider various aspects of the Schur problem for a square complex matrix <i>A</i>, namely the similarity unitary transformation of <i>A</i> into upper triangular form containing the eigenvalues of <i>A</i> 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 <i>A</i>. 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. |
| format | Article |
| id | doaj-art-2201328c8e6f492aa7ee0f21fa8b0c2e |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-2201328c8e6f492aa7ee0f21fa8b0c2e2024-12-27T14:10:22ZengMDPI AGAxioms2075-16802024-11-01131283910.3390/axioms13120839On Schur Forms for Matrices with Simple EigenvaluesMihail Mihaylov Konstantinov0Petko Hristov Petkov1Department of Mathematics, University of Architecture, Civil Engineering and Geodesy, 1064 Sofia, BulgariaBulgarian Academy of Sciences, 1040 Sofia, BulgariaIn this paper, we consider various aspects of the Schur problem for a square complex matrix <i>A</i>, namely the similarity unitary transformation of <i>A</i> into upper triangular form containing the eigenvalues of <i>A</i> 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 <i>A</i>. 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.https://www.mdpi.com/2075-1680/13/12/839Schur canonical formSchur condensed formdiagonally spectral matrixquasi-Schur formperturbations of Schur form |
| spellingShingle | Mihail Mihaylov Konstantinov Petko Hristov Petkov On Schur Forms for Matrices with Simple Eigenvalues Axioms Schur canonical form Schur condensed form diagonally spectral matrix quasi-Schur form perturbations of Schur form |
| title | On Schur Forms for Matrices with Simple Eigenvalues |
| title_full | On Schur Forms for Matrices with Simple Eigenvalues |
| title_fullStr | On Schur Forms for Matrices with Simple Eigenvalues |
| title_full_unstemmed | On Schur Forms for Matrices with Simple Eigenvalues |
| title_short | On Schur Forms for Matrices with Simple Eigenvalues |
| title_sort | on schur forms for matrices with simple eigenvalues |
| topic | Schur canonical form Schur condensed form diagonally spectral matrix quasi-Schur form perturbations of Schur form |
| url | https://www.mdpi.com/2075-1680/13/12/839 |
| work_keys_str_mv | AT mihailmihaylovkonstantinov onschurformsformatriceswithsimpleeigenvalues AT petkohristovpetkov onschurformsformatriceswithsimpleeigenvalues |