Singular values of differences of positive semidefinite matrices

Xingzhi Zhan

doi:10.1137/S0895479800369840

Singular values of differences of positive semidefinite matrices

Xingzhi Zhan^*

^*Corresponding author for this work

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

74 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	819-823
Number of pages	5
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	22
Issue number	3
DOIs	https://doi.org/10.1137/S0895479800369840
State	Published - Oct 2000
Externally published	Yes

Keywords

Majorization
Positive semidefinite matrices
Singular values
Unitarily invariant norms

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title = "Singular values of differences of positive semidefinite matrices",

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",

year = "2000",

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doi = "10.1137/S0895479800369840",

language = "英语",

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TY - JOUR

T1 - Singular values of differences of positive semidefinite matrices

AU - Zhan, Xingzhi

PY - 2000/10

Y1 - 2000/10

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

UR - https://www.scopus.com/pages/publications/0035219521

U2 - 10.1137/S0895479800369840

DO - 10.1137/S0895479800369840

M3 - 文章

AN - SCOPUS:0035219521

SN - 0895-4798

VL - 22

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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