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Simultaneous block diagonalization of symmetric matrices via congruence.

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Title: Simultaneous block diagonalization of symmetric matrices via congruence.
Authors: Fang, Lishan1 (AUTHOR), Huang, Hua-Lin1 (AUTHOR) hualin.huang@hqu.edu.cn, Huang, Jiayan1 (AUTHOR)
Source: Linear Algebra & its Applications. May2026, Vol. 737, p193-212. 20p.
Subjects: Symmetric matrices, Matrix decomposition, Mathematical equivalence, Idempotents, Normal forms (Mathematics), Matrices (Mathematics)
Abstract: This article studies canonical forms derived from the finest simultaneous block diagonalization of a set of symmetric matrices via congruence. Our technique relies on the center theory of a set of multivariate polynomials, which we adapt for a set of symmetric matrices. We establish a bijective relationship between the simultaneous block diagonalizations of these matrices via congruence and complete sets of orthogonal idempotents of their centers. Based on this framework, we provide an algorithm relying primarily on standard linear algebra tasks. We also extend this technique to the simultaneous block diagonalizations of a set of Hermitian matrices via ⁎-congruence. In addition, we investigate the simultaneous orthogonal block diagonalizations for real matrices and the simultaneous unitary block diagonalizations for Hermitian matrices. Several examples are provided to demonstrate its effectiveness. [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: 192196634
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
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  Data: Simultaneous block diagonalization of symmetric matrices via congruence.
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  Data: <searchLink fieldCode="AR" term="%22Fang%2C+Lishan%22">Fang, Lishan</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Huang%2C+Hua-Lin%22">Huang, Hua-Lin</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> hualin.huang@hqu.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Huang%2C+Jiayan%22">Huang, Jiayan</searchLink><relatesTo>1</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22Linear+Algebra+%26+its+Applications%22">Linear Algebra & its Applications</searchLink>. May2026, Vol. 737, p193-212. 20p.
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  Data: <searchLink fieldCode="DE" term="%22Symmetric+matrices%22">Symmetric matrices</searchLink><br /><searchLink fieldCode="DE" term="%22Matrix+decomposition%22">Matrix decomposition</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+equivalence%22">Mathematical equivalence</searchLink><br /><searchLink fieldCode="DE" term="%22Idempotents%22">Idempotents</searchLink><br /><searchLink fieldCode="DE" term="%22Normal+forms+%28Mathematics%29%22">Normal forms (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Matrices+%28Mathematics%29%22">Matrices (Mathematics)</searchLink>
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  Label: Abstract
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  Data: This article studies canonical forms derived from the finest simultaneous block diagonalization of a set of symmetric matrices via congruence. Our technique relies on the center theory of a set of multivariate polynomials, which we adapt for a set of symmetric matrices. We establish a bijective relationship between the simultaneous block diagonalizations of these matrices via congruence and complete sets of orthogonal idempotents of their centers. Based on this framework, we provide an algorithm relying primarily on standard linear algebra tasks. We also extend this technique to the simultaneous block diagonalizations of a set of Hermitian matrices via ⁎-congruence. In addition, we investigate the simultaneous orthogonal block diagonalizations for real matrices and the simultaneous unitary block diagonalizations for Hermitian matrices. Several examples are provided to demonstrate its effectiveness. [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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RecordInfo BibRecord:
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      – Type: doi
        Value: 10.1016/j.laa.2026.02.016
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      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 20
        StartPage: 193
    Subjects:
      – SubjectFull: Symmetric matrices
        Type: general
      – SubjectFull: Matrix decomposition
        Type: general
      – SubjectFull: Mathematical equivalence
        Type: general
      – SubjectFull: Idempotents
        Type: general
      – SubjectFull: Normal forms (Mathematics)
        Type: general
      – SubjectFull: Matrices (Mathematics)
        Type: general
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      – TitleFull: Simultaneous block diagonalization of symmetric matrices via congruence.
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            NameFull: Fang, Lishan
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            NameFull: Huang, Hua-Lin
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            NameFull: Huang, Jiayan
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            – D: 15
              M: 05
              Text: May2026
              Type: published
              Y: 2026
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              Value: 737
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