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On normal forms for the similarity classes of matrices and pairs of matrices.

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Title: On normal forms for the similarity classes of matrices and pairs of matrices.
Authors: Bongartz, Klaus1 (AUTHOR) klausbongartz@online.de
Source: Linear Algebra & its Applications. Aug2026, Vol. 743, p76-86. 11p.
Subjects: Normal forms (Mathematics), Linear operators
Abstract: We answer two questions of Kontsevich posed 1998 in the book 'Arnold's problems'. First, over any field k there is a representative system for the similarity classes of n × n -matrices which is a finite disjoint union of affine subspaces. And second, for n ≥ 2 an analogous statement fails for pairs of n × n -matrices over any algebraically closed field of characteristic 0. The positive answer rests on a slight modification of the rational normal form whereas the negative answer requires more work. [ABSTRACT FROM AUTHOR]
Copyright of Linear Algebra & its Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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DbLabel: Engineering Source
An: 193660180
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PubType: Academic Journal
PubTypeId: academicJournal
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  Data: On normal forms for the similarity classes of matrices and pairs of matrices.
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  Label: Abstract
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  Data: We answer two questions of Kontsevich posed 1998 in the book 'Arnold's problems'. First, over any field k there is a representative system for the similarity classes of n × n -matrices which is a finite disjoint union of affine subspaces. And second, for n ≥ 2 an analogous statement fails for pairs of n × n -matrices over any algebraically closed field of characteristic 0. The positive answer rests on a slight modification of the rational normal form whereas the negative answer requires more work. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Linear Algebra & its Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1016/j.laa.2026.04.017
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      – Code: eng
        Text: English
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      Pagination:
        PageCount: 11
        StartPage: 76
    Subjects:
      – SubjectFull: Normal forms (Mathematics)
        Type: general
      – SubjectFull: Linear operators
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      – TitleFull: On normal forms for the similarity classes of matrices and pairs of matrices.
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              Text: Aug2026
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