Inequalities for unitarily invariant norms

Xingzhi Zhan

doi:10.1137/S0895479898323823

Inequalities for unitarily invariant norms

Xingzhi Zhan

Peking University

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

59 Scopus citations

Abstract

Let A, B, X be complex matrices with A, B positive semidefinite. It is proved that (2 + t)∥A^r X B^2-r + A^2-r X B^r ∥ ≤ 2∥A² X + tAXB + X B²∥ for any unitarily invariant norm ∥ · ∥ and real numbers r, t satisfying 1 ≤ 2r ≤ 3, -2 < t ≤ 2. The case r = 1, t = 0 of this result is the well-known arithmetic-geometric mean inequality due to R. Bhatia and C. Davis [SIAM J. Matrix Anal. Appl., 14 (1993), pp. 132-136]. Several other unitarily invariant norm inequalities are derived.

Original language	English
Pages (from-to)	466-470
Number of pages	5
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	20
Issue number	2
DOIs	https://doi.org/10.1137/S0895479898323823
State	Published - 1998
Externally published	Yes

Keywords

Arithmetic-geometric mean inequality
Hadamard product
Unitarily invariant norm

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abstract = "Let A, B, X be complex matrices with A, B positive semidefinite. It is proved that (2 + t)∥Ar X B2-r + A2-r X Br ∥ ≤ 2∥A2 X + tAXB + X B2∥ for any unitarily invariant norm ∥ · ∥ and real numbers r, t satisfying 1 ≤ 2r ≤ 3, -2 < t ≤ 2. The case r = 1, t = 0 of this result is the well-known arithmetic-geometric mean inequality due to R. Bhatia and C. Davis [SIAM J. Matrix Anal. Appl., 14 (1993), pp. 132-136]. Several other unitarily invariant norm inequalities are derived.",

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T1 - Inequalities for unitarily invariant norms

AU - Zhan, Xingzhi

PY - 1998

Y1 - 1998

N2 - Let A, B, X be complex matrices with A, B positive semidefinite. It is proved that (2 + t)∥Ar X B2-r + A2-r X Br ∥ ≤ 2∥A2 X + tAXB + X B2∥ for any unitarily invariant norm ∥ · ∥ and real numbers r, t satisfying 1 ≤ 2r ≤ 3, -2 < t ≤ 2. The case r = 1, t = 0 of this result is the well-known arithmetic-geometric mean inequality due to R. Bhatia and C. Davis [SIAM J. Matrix Anal. Appl., 14 (1993), pp. 132-136]. Several other unitarily invariant norm inequalities are derived.

AB - Let A, B, X be complex matrices with A, B positive semidefinite. It is proved that (2 + t)∥Ar X B2-r + A2-r X Br ∥ ≤ 2∥A2 X + tAXB + X B2∥ for any unitarily invariant norm ∥ · ∥ and real numbers r, t satisfying 1 ≤ 2r ≤ 3, -2 < t ≤ 2. The case r = 1, t = 0 of this result is the well-known arithmetic-geometric mean inequality due to R. Bhatia and C. Davis [SIAM J. Matrix Anal. Appl., 14 (1993), pp. 132-136]. Several other unitarily invariant norm inequalities are derived.

KW - Arithmetic-geometric mean inequality

KW - Hadamard product

KW - Unitarily invariant norm

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DO - 10.1137/S0895479898323823

M3 - 文章

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SN - 0895-4798

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SP - 466

EP - 470

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 2

ER -

Inequalities for unitarily invariant norms

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Keywords

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