Singular values of differences of positive semidefinite matrices

Xingzhi Zhan

doi:10.1137/S0895479800369840

Singular values of differences of positive semidefinite matrices

Xingzhi Zhan^*

^*此作品的通讯作者

Peking University
Tohoku University

科研成果: 期刊稿件 › 文章 › 同行评审

78 引用（Scopus）

摘要

Let M_n be the space of n × n complex matrices. For A ∈ M_n, let s(A) ≡ (s₁(A), . . . , S_n(A)), where s₁(A) ≥ ⋯ ≥ S_n(A) are the singular values of A. We prove that if A, B ∈ M_n are positive semidefinite, then (i) s_j(A - B) ≤ s_j(A ⊕ B), j = 1,2, . . . , n, and (ii) the weak log-majorization relations s(A-|z|B) ≺_wlog s(A+zB) ≺_wlog s(A + \z\B) hold for any complex number z. This sharpens some results due to R. Bhatia and F. Kittaneh.

源语言	英语
页（从-至）	819-823
页数	5
期刊	SIAM Journal on Matrix Analysis and Applications
卷	22
期	3
DOI	https://doi.org/10.1137/S0895479800369840
出版状态	已出版 - 10月 2000
已对外发布	是

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abstract = "Let Mn be the space of n × n complex matrices. For A ∈ Mn, let s(A) ≡ (s1(A), . . . , Sn(A)), where s1(A) ≥ ⋯ ≥ Sn(A) are the singular values of A. We prove that if A, B ∈ Mn are positive semidefinite, then (i) sj(A - B) ≤ sj(A ⊕ B), j = 1,2, . . . , n, and (ii) the weak log-majorization relations s(A-|z|B) ≺wlog s(A+zB) ≺wlog s(A + \textbackslash{}z\textbackslash{}B) hold for any complex number z. This sharpens some results due to R. Bhatia and F. Kittaneh.",

keywords = "Majorization, Positive semidefinite matrices, Singular values, Unitarily invariant norms",

author = "Xingzhi Zhan",

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T1 - Singular values of differences of positive semidefinite matrices

AU - Zhan, Xingzhi

PY - 2000/10

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N2 - Let Mn be the space of n × n complex matrices. For A ∈ Mn, let s(A) ≡ (s1(A), . . . , Sn(A)), where s1(A) ≥ ⋯ ≥ Sn(A) are the singular values of A. We prove that if A, B ∈ Mn are positive semidefinite, then (i) sj(A - B) ≤ sj(A ⊕ B), j = 1,2, . . . , n, and (ii) the weak log-majorization relations s(A-|z|B) ≺wlog s(A+zB) ≺wlog s(A + \z\B) hold for any complex number z. This sharpens some results due to R. Bhatia and F. Kittaneh.

AB - Let Mn be the space of n × n complex matrices. For A ∈ Mn, let s(A) ≡ (s1(A), . . . , Sn(A)), where s1(A) ≥ ⋯ ≥ Sn(A) are the singular values of A. We prove that if A, B ∈ Mn are positive semidefinite, then (i) sj(A - B) ≤ sj(A ⊕ B), j = 1,2, . . . , n, and (ii) the weak log-majorization relations s(A-|z|B) ≺wlog s(A+zB) ≺wlog s(A + \z\B) hold for any complex number z. This sharpens some results due to R. Bhatia and F. Kittaneh.

KW - Majorization

KW - Positive semidefinite matrices

KW - Singular values

KW - Unitarily invariant norms

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DO - 10.1137/S0895479800369840

M3 - 文章

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SN - 0895-4798

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SP - 819

EP - 823

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 3

ER -

Singular values of differences of positive semidefinite matrices

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