Inequalities for the singular values of Hadamard products

Xingzhi Zhan

doi:10.1137/S0895479896309645

Inequalities for the singular values of Hadamard products

Xingzhi Zhan^*

^*Corresponding author for this work

Peking University

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

10 Scopus citations

Abstract

Let M_m,n be the space of m X n complex matrices. For A, B ∈ M_m,n, denote the Hadamard (or Schur) product of A and B by A ○ B. Given A ∈ M_m,n, let σ₁(A) ≥ σ₂(A) ≥ ⋯ ≥ σ_min{m,n}(A) be the ordered singular values, and the decreasingly ordered Euclidean row and column lengths of A are denoted by r₁(A) ≥ r₂(A) ≥ ⋯ ≥ r_m(A) and c₁(A) ≥ c₂(A) ≥ ⋯ ≥ c_n(A), respectively. It is shown that for any A, B ∈ M_m,n, (equation presented) k = 1, 2, . . . , min{m, n}. This settles, in a stronger form, a conjecture of R. A. Horn and C. R. Johnson [Topics in Matrix Analysis, Cambridge University Press, New York, 1991, p. 344] affirmatively.

Original language	English
Pages (from-to)	1093-1095
Number of pages	3
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	18
Issue number	4
DOIs	https://doi.org/10.1137/S0895479896309645
State	Published - Oct 1997
Externally published	Yes

Keywords

Hadamard products
Singular values

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10.1137/S0895479896309645

Cite this

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title = "Inequalities for the singular values of Hadamard products",

abstract = "Let Mm,n be the space of m X n complex matrices. For A, B ∈ Mm,n, denote the Hadamard (or Schur) product of A and B by A ○ B. Given A ∈ Mm,n, let σ1(A) ≥ σ2(A) ≥ ⋯ ≥ σmin\{m,n\}(A) be the ordered singular values, and the decreasingly ordered Euclidean row and column lengths of A are denoted by r1(A) ≥ r2(A) ≥ ⋯ ≥ rm(A) and c1(A) ≥ c2(A) ≥ ⋯ ≥ cn(A), respectively. It is shown that for any A, B ∈ Mm,n, (equation presented) k = 1, 2, . . . , min\{m, n\}. This settles, in a stronger form, a conjecture of R. A. Horn and C. R. Johnson [Topics in Matrix Analysis, Cambridge University Press, New York, 1991, p. 344] affirmatively.",

keywords = "Hadamard products, Singular values",

author = "Xingzhi Zhan",

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T1 - Inequalities for the singular values of Hadamard products

AU - Zhan, Xingzhi

PY - 1997/10

Y1 - 1997/10

N2 - Let Mm,n be the space of m X n complex matrices. For A, B ∈ Mm,n, denote the Hadamard (or Schur) product of A and B by A ○ B. Given A ∈ Mm,n, let σ1(A) ≥ σ2(A) ≥ ⋯ ≥ σmin{m,n}(A) be the ordered singular values, and the decreasingly ordered Euclidean row and column lengths of A are denoted by r1(A) ≥ r2(A) ≥ ⋯ ≥ rm(A) and c1(A) ≥ c2(A) ≥ ⋯ ≥ cn(A), respectively. It is shown that for any A, B ∈ Mm,n, (equation presented) k = 1, 2, . . . , min{m, n}. This settles, in a stronger form, a conjecture of R. A. Horn and C. R. Johnson [Topics in Matrix Analysis, Cambridge University Press, New York, 1991, p. 344] affirmatively.

AB - Let Mm,n be the space of m X n complex matrices. For A, B ∈ Mm,n, denote the Hadamard (or Schur) product of A and B by A ○ B. Given A ∈ Mm,n, let σ1(A) ≥ σ2(A) ≥ ⋯ ≥ σmin{m,n}(A) be the ordered singular values, and the decreasingly ordered Euclidean row and column lengths of A are denoted by r1(A) ≥ r2(A) ≥ ⋯ ≥ rm(A) and c1(A) ≥ c2(A) ≥ ⋯ ≥ cn(A), respectively. It is shown that for any A, B ∈ Mm,n, (equation presented) k = 1, 2, . . . , min{m, n}. This settles, in a stronger form, a conjecture of R. A. Horn and C. R. Johnson [Topics in Matrix Analysis, Cambridge University Press, New York, 1991, p. 344] affirmatively.

KW - Hadamard products

KW - Singular values

UR - https://www.scopus.com/pages/publications/0031518874

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DO - 10.1137/S0895479896309645

M3 - 文章

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SN - 0895-4798

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JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 4

ER -

Inequalities for the singular values of Hadamard products

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Keywords

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