Norm inequalities for cartesian decompositions

Xingzhi Zhan

doi:10.1016/S0024-3795(98)10174-X

Norm inequalities for cartesian decompositions

Xingzhi Zhan^*

^*Corresponding author for this work

Peking University

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

Let the Cartesian decomposition of a complex n × n matrix T be T = A + iB with A, B Hermitian. Let α_j and β_j be the eigenvalues of A and B respectively ordered so that |α₁| ≥ ⋯ ≥ |α_n| and |β₁| ≥ ⋯ ≥ |β_n|. We prove that ∥diag(α₁ + iβ₁,⋯, α_n + iβ_n)∥ ≤ √2∥Τ∥ for every unitarily invariant norm this settles affirmatively a conjecture of Ando and Bhatia (T. Ando, R. Bhatia, Eigenvalue inequalities associated with the cartesian decomposition, Linear and Multilinear Algebra 22 (1987) 133).

Original language	English
Pages (from-to)	297-301
Number of pages	5
Journal	Linear Algebra and Its Applications
Volume	286
Issue number	1-3
DOIs	https://doi.org/10.1016/S0024-3795(98)10174-X
State	Published - 1 Jan 1999
Externally published	Yes

Keywords

Cartesian decomposition
Eigenvalue
Singular value
Unitarily invariant norm

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10.1016/S0024-3795(98)10174-X

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title = "Norm inequalities for cartesian decompositions",

abstract = "Let the Cartesian decomposition of a complex n × n matrix T be T = A + iB with A, B Hermitian. Let αj and βj be the eigenvalues of A and B respectively ordered so that |α1| ≥ ⋯ ≥ |αn| and |β1| ≥ ⋯ ≥ |βn|. We prove that ∥diag(α1 + iβ1,⋯, αn + iβn)∥ ≤ √2∥Τ∥ for every unitarily invariant norm this settles affirmatively a conjecture of Ando and Bhatia (T. Ando, R. Bhatia, Eigenvalue inequalities associated with the cartesian decomposition, Linear and Multilinear Algebra 22 (1987) 133).",

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T1 - Norm inequalities for cartesian decompositions

AU - Zhan, Xingzhi

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N2 - Let the Cartesian decomposition of a complex n × n matrix T be T = A + iB with A, B Hermitian. Let αj and βj be the eigenvalues of A and B respectively ordered so that |α1| ≥ ⋯ ≥ |αn| and |β1| ≥ ⋯ ≥ |βn|. We prove that ∥diag(α1 + iβ1,⋯, αn + iβn)∥ ≤ √2∥Τ∥ for every unitarily invariant norm this settles affirmatively a conjecture of Ando and Bhatia (T. Ando, R. Bhatia, Eigenvalue inequalities associated with the cartesian decomposition, Linear and Multilinear Algebra 22 (1987) 133).

AB - Let the Cartesian decomposition of a complex n × n matrix T be T = A + iB with A, B Hermitian. Let αj and βj be the eigenvalues of A and B respectively ordered so that |α1| ≥ ⋯ ≥ |αn| and |β1| ≥ ⋯ ≥ |βn|. We prove that ∥diag(α1 + iβ1,⋯, αn + iβn)∥ ≤ √2∥Τ∥ for every unitarily invariant norm this settles affirmatively a conjecture of Ando and Bhatia (T. Ando, R. Bhatia, Eigenvalue inequalities associated with the cartesian decomposition, Linear and Multilinear Algebra 22 (1987) 133).

KW - Cartesian decomposition

KW - Eigenvalue

KW - Singular value

KW - Unitarily invariant norm

UR - https://www.scopus.com/pages/publications/0040219794

U2 - 10.1016/S0024-3795(98)10174-X

DO - 10.1016/S0024-3795(98)10174-X

M3 - 文章

AN - SCOPUS:0040219794

SN - 0024-3795

VL - 286

SP - 297

EP - 301

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 1-3

ER -

Norm inequalities for cartesian decompositions

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Keywords

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