Multi-variable weighted geometric means of positive definite matrices

Hosoo Lee; Yongdo Lim; Takeaki Yamazaki

doi:10.1016/j.laa.2011.01.026

Multi-variable weighted geometric means of positive definite matrices

Hosoo Lee, Yongdo Lim, Takeaki Yamazaki

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

18 Scopus citations

Abstract

We define a family of weighted geometric means {G(t;ωA)}t∈[0,1] ⁿ where ω and A vary over all positive probability vectors in ^Rn and n-tuples of positive definite matrices resp. Each of these weighted geometric means interpolates between the weighted ALM (t=0_n) and BMP (t=1_n) geometric means (ALM and BMP geometric means have been defined by Ando-Li-Mathias and Bini-Meini-Poloni, respectively.) We show that the weighted geometric means satisfy multidimensional versions of all properties that one would expect for a two-variable weighted geometric mean.

Original language	English
Pages (from-to)	307-322
Number of pages	16
Journal	Linear Algebra and Its Applications
Volume	435
Issue number	2
DOIs	https://doi.org/10.1016/j.laa.2011.01.026
State	Published - 15 Jul 2011
Externally published	Yes

Keywords

ALM geometric mean
Elementary symmetric polynomial
Positive definite operators
Thompson metric
Weighted geometric mean

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10.1016/j.laa.2011.01.026

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title = "Multi-variable weighted geometric means of positive definite matrices",

abstract = "We define a family of weighted geometric means \{G(t;ωA)\}t∈[0,1] n where ω and A vary over all positive probability vectors in Rn and n-tuples of positive definite matrices resp. Each of these weighted geometric means interpolates between the weighted ALM (t=0n) and BMP (t=1n) geometric means (ALM and BMP geometric means have been defined by Ando-Li-Mathias and Bini-Meini-Poloni, respectively.) We show that the weighted geometric means satisfy multidimensional versions of all properties that one would expect for a two-variable weighted geometric mean.",

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T1 - Multi-variable weighted geometric means of positive definite matrices

AU - Lee, Hosoo

AU - Lim, Yongdo

AU - Yamazaki, Takeaki

PY - 2011/7/15

Y1 - 2011/7/15

N2 - We define a family of weighted geometric means {G(t;ωA)}t∈[0,1] n where ω and A vary over all positive probability vectors in Rn and n-tuples of positive definite matrices resp. Each of these weighted geometric means interpolates between the weighted ALM (t=0n) and BMP (t=1n) geometric means (ALM and BMP geometric means have been defined by Ando-Li-Mathias and Bini-Meini-Poloni, respectively.) We show that the weighted geometric means satisfy multidimensional versions of all properties that one would expect for a two-variable weighted geometric mean.

AB - We define a family of weighted geometric means {G(t;ωA)}t∈[0,1] n where ω and A vary over all positive probability vectors in Rn and n-tuples of positive definite matrices resp. Each of these weighted geometric means interpolates between the weighted ALM (t=0n) and BMP (t=1n) geometric means (ALM and BMP geometric means have been defined by Ando-Li-Mathias and Bini-Meini-Poloni, respectively.) We show that the weighted geometric means satisfy multidimensional versions of all properties that one would expect for a two-variable weighted geometric mean.

KW - ALM geometric mean

KW - Elementary symmetric polynomial

KW - Positive definite operators

KW - Thompson metric

KW - Weighted geometric mean

UR - https://www.scopus.com/pages/publications/79952748045

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DO - 10.1016/j.laa.2011.01.026

M3 - Article

AN - SCOPUS:79952748045

SN - 0024-3795

VL - 435

SP - 307

EP - 322

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 2

ER -

Multi-variable weighted geometric means of positive definite matrices

Abstract

Keywords

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