{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:37:45Z","timestamp":1760146665699,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,11,28]],"date-time":"2024-11-28T00:00:00Z","timestamp":1732752000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/axioms13120839","type":"journal-article","created":{"date-parts":[[2024,11,28]],"date-time":"2024-11-28T08:15:54Z","timestamp":1732781754000},"page":"839","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Schur Forms for Matrices with Simple Eigenvalues"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3708-516X","authenticated-orcid":false,"given":"Mihail Mihaylov","family":"Konstantinov","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Architecture, Civil Engineering and Geodesy, 1064 Sofia, Bulgaria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0334-0900","authenticated-orcid":false,"given":"Petko Hristov","family":"Petkov","sequence":"additional","affiliation":[{"name":"Bulgarian Academy of Sciences, 1040 Sofia, Bulgaria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,28]]},"reference":[{"key":"ref_1","first-page":"159","article-title":"Beitr\u00e4ge zur Theorie der Gruppen linearer homogener Substitutionen","volume":"10","author":"Schur","year":"1909","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Bhatia, R. (1996). Matrix Analysis, Springer.","DOI":"10.1007\/978-1-4612-0653-8"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Golub, G., and Loan, C.V. (2013). Matrix Computations, The Johns Hopkins University Press. [4th ed.].","DOI":"10.56021\/9781421407944"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Horn, R., and Johnson, C. (2012). Matrix Analysis, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9781139020411"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Horn, R., and Johnson, C. (1991). Topics in Matrix Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511840371"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1137\/S089547989120267X","article-title":"Nonlocal perturbation analysis of the Schur system of a matrix","volume":"15","author":"Konstantinov","year":"1994","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Konstantinov, M., and Petkov, P. (2020). Perturbation Methods in Matrix Analysis and Control, Nova Science Publishers.","DOI":"10.52305\/EZCI5944"},{"key":"ref_8","unstructured":"Minekova, A., Nitch-Griffin, E., and Olshevsky, V. (2021). Gohberg-Kaashoek numbers and forward stability of Schur canonical forms. arXiv."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Minekova, A., Nitch-Griffin, E., and Olshevsky, V. (2024). Backward stability of the Schur decomposition under small perturbations. 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Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1007\/s00211-006-0043-0","article-title":"A numerical method for computing the Hamiltonian Schur form","volume":"105","author":"Chu","year":"2007","journal-title":"Numer. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"126446","DOI":"10.1016\/j.neucom.2023.126359","article-title":"Perturbations of Tensor-Schur decomposition and its applications to multivariable control systems and facial recognitions","volume":"547","author":"Chen","year":"2023","journal-title":"Neurocomputing"},{"key":"ref_15","unstructured":"Stewart, G., and Sun, J. (1990). Matrix Perturbation Theory, Academic Press."},{"key":"ref_16","unstructured":"Konstantinov, M., Gu, D., Mehrmann, V., and Petkov, P. (2003). Perturbation Theory for Matrix Equations, Science Direct."},{"key":"ref_17","first-page":"413","article-title":"Invariants and canonical forms for linear multivariable systems under the action of orthogonal transformation groups","volume":"17","author":"Konstantinov","year":"1981","journal-title":"Kybernetika"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/0024-3795(95)00647-8","article-title":"Perturbation analysis of orthogonal canonical forms","volume":"251","author":"Konstantinov","year":"1997","journal-title":"Linear Algebra Appl."},{"key":"ref_19","unstructured":"Byrnes, C. (1997). Numerical Methods for Linear Control Systems. Systems and Control in the Twenty-First Century. Systems & Control: Foundations & Applications, Birkh\u00e4user."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1328","DOI":"10.1137\/S0895479892242189","article-title":"Perturbation bounds for the generalized Schur decomposition","volume":"16","author":"Sun","year":"1995","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1016\/0024-3795(95)00107-7","article-title":"Perturbation analysis of system Hessenberg and Hessenberg\/triangular forms","volume":"241\/243","author":"Sun","year":"1996","journal-title":"Linear Algebra Appl."},{"key":"ref_22","unstructured":"The MathWorks, Inc (2020). MATLAB Version 9.9.0.1538559 (R2020b), The MathWorks, Inc."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1073\/pnas.17.7.417","article-title":"A canonical form for real matrices under orthogonal transformations","volume":"17","author":"Murnaghan","year":"1931","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/0024-3795(91)90232-L","article-title":"A survey of canonical forms and invariants for unitary similarity","volume":"147","author":"Shapiro","year":"1991","journal-title":"Linear Algebra Appl."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Higham, N. (2008). Functions of Matrices: Theory and Computation, SIAM.","DOI":"10.1137\/1.9780898717778"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/BF02392670","article-title":"The problem of unitary equivalence","volume":"86","author":"Brenner","year":"1951","journal-title":"Acta Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1112\/jlms\/s1-28.3.314","article-title":"On unitary equivalence","volume":"28","author":"Littlewood","year":"1953","journal-title":"J. Lond. Math. Soc."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1134\/S1064562408030101","article-title":"The canonical Schur form of a matrix with simple eigenvalues","volume":"77","author":"Ikramov","year":"2008","journal-title":"Dokl. Math."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Hartshorne, R. (1977). Algebraic Geometry, Springer.","DOI":"10.1007\/978-1-4757-3849-0"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1134\/S0965542510030012","article-title":"On a constructive procedure for verifying whether a matrix can be made real by a unitary similarity transformation","volume":"50","author":"Ikramov","year":"2010","journal-title":"Comput. Math. Math. Phys."},{"key":"ref_31","unstructured":"(2019). IEEE Standard for Floating-Point Arithmetic (Standard No. 754-2019)."},{"key":"ref_32","unstructured":"Glazman, M., and Ljubich, J. (2006). Finite-Dimensional Linear Analysis: A Systematic Presentation in Problem Form (Dover Books in Mathematics), Dover Publications."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"893","DOI":"10.2140\/pjm.1959.9.893","article-title":"On the similarity transformation between a matrix and its transpose","volume":"9","author":"Tausky","year":"1959","journal-title":"Pac. J. Math."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Arnold, V. (1988). 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