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From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues

    Authors

    • Kh.D. Ikramov
    • Alimohammad Nazari

    aMoscow Lomonosov State University, Moscow, Russia

,

Document Type : Regular paper

10.52547/CMCMA.1.1.17
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Abstract

Williamson's theorem states that every real symmetric positive definite matrix $A$ of even order can be brought to diagonal form via a symplectic $T$-congruence transformation. The diagonal entries of the resulting diagonal form are called the symplectic eigenvalues of $A$. We point at an analog of this classical result related to Hermitian positive definite matrices, *-congruences, and another class of transformation matrices, namely, pseudo-unitary matrices. This leads to the concept of pseudo-unitary (or pseudo-orthogonal, in the real case) eigenvalues of positive definite matrices.

Keywords

  • congruence transformation
  • symplectic matrix
  • pseudo-unitary matrix
  • indices of inertia
  • Schur inequality
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Computational Mathematics and Computer Modeling with Applications (CMCMA)
Volume 1, Issue 1 - Serial Number 1
June 2022
Pages 17-20
Files
  • XML
  • PDF 156.56 K
History
  • Receive Date: 23 December 2021
  • Revise Date: 27 December 2021
  • Accept Date: 31 December 2021
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How to cite
  • RIS
  • EndNote
  • Mendeley
  • BibTeX
  • APA
  • MLA
  • HARVARD
  • CHICAGO
  • VANCOUVER
Statistics
  • Article View: 308
  • PDF Download: 460

APA

Ikramov, K. and Nazari, A. (2022). From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues. Computational Mathematics and Computer Modeling with Applications (CMCMA), 1(1), 17-20. doi: 10.52547/CMCMA.1.1.17

MLA

Ikramov, K. , and Nazari, A. . "From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues", Computational Mathematics and Computer Modeling with Applications (CMCMA), 1, 1, 2022, 17-20. doi: 10.52547/CMCMA.1.1.17

HARVARD

Ikramov, K., Nazari, A. (2022). 'From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues', Computational Mathematics and Computer Modeling with Applications (CMCMA), 1(1), pp. 17-20. doi: 10.52547/CMCMA.1.1.17

CHICAGO

K. Ikramov and A. Nazari, "From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues," Computational Mathematics and Computer Modeling with Applications (CMCMA), 1 1 (2022): 17-20, doi: 10.52547/CMCMA.1.1.17

VANCOUVER

Ikramov, K., Nazari, A. From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues. Computational Mathematics and Computer Modeling with Applications (CMCMA), 2022; 1(1): 17-20. doi: 10.52547/CMCMA.1.1.17

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