Reciprocal Lie-Trotter formula

Koenraad Audenaert; F. Hiai

doi:10.1080/03081087.2015.1082957

Reciprocal Lie-Trotter formula

Koenraad Audenaert, F. Hiai

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

Abstract

Let_A and _B be positive semidefinite matrices. The limit of the expression Z_p := (A^p/2B^p A^p/2)^1/p as _p tends to ₀ is given by the well-known Lie–Trotter formula. A similar formula holds for the limit of G_p := (A^p # B^p)^2/pas _p tends to ₀, where _{X # Y}is the geometric mean of _X and _Y. In this paper we study the limit of Z_p and G_p as _p tends to _∞ instead of ₀, with the ultimate goal of finding an explicit formula, which we call the reciprocal Lie–Trotter formula. We show that the limit of Z_p exists and find an explicit formula in a special case. The limit of G_p is shown for _{2 × 2} matrices only.

Original language	English
Pages (from-to)	1220-1235
Number of pages	16
Journal	Linear and Multilinear Algebra
Volume	64
Issue number	6
Early online date	7 Sept 2015
DOIs	https://doi.org/10.1080/03081087.2015.1082957
Publication status	Published - 2016

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Cite this

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title = "Reciprocal Lie-Trotter formula",

abstract = "Let A and B be positive semidefinite matrices. The limit of the expression Zp := (Ap/2 Bp Ap/2)1/p as p tends to 0 is given by the well-known Lie–Trotter formula. A similar formula holds for the limit of Gp := (Ap # Bp)2/p as p tends to 0, where X # Y is the geometric mean of X and Y. In this paper we study the limit of Zp and Gp as p tends to ∞ instead of 0, with the ultimate goal of finding an explicit formula, which we call the reciprocal Lie–Trotter formula. We show that the limit of Zp exists and find an explicit formula in a special case. The limit of Gp is shown for 2 × 2 matrices only.",

author = "Koenraad Audenaert and F. Hiai",

year = "2016",

doi = "10.1080/03081087.2015.1082957",

language = "English",

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journal = "Linear and Multilinear Algebra",

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T1 - Reciprocal Lie-Trotter formula

AU - Audenaert, Koenraad

AU - Hiai, F.

PY - 2016

Y1 - 2016

N2 - Let A and B be positive semidefinite matrices. The limit of the expression Zp := (Ap/2 Bp Ap/2)1/p as p tends to 0 is given by the well-known Lie–Trotter formula. A similar formula holds for the limit of Gp := (Ap # Bp)2/p as p tends to 0, where X # Y is the geometric mean of X and Y. In this paper we study the limit of Zp and Gp as p tends to ∞ instead of 0, with the ultimate goal of finding an explicit formula, which we call the reciprocal Lie–Trotter formula. We show that the limit of Zp exists and find an explicit formula in a special case. The limit of Gp is shown for 2 × 2 matrices only.

AB - Let A and B be positive semidefinite matrices. The limit of the expression Zp := (Ap/2 Bp Ap/2)1/p as p tends to 0 is given by the well-known Lie–Trotter formula. A similar formula holds for the limit of Gp := (Ap # Bp)2/p as p tends to 0, where X # Y is the geometric mean of X and Y. In this paper we study the limit of Zp and Gp as p tends to ∞ instead of 0, with the ultimate goal of finding an explicit formula, which we call the reciprocal Lie–Trotter formula. We show that the limit of Zp exists and find an explicit formula in a special case. The limit of Gp is shown for 2 × 2 matrices only.

U2 - 10.1080/03081087.2015.1082957

DO - 10.1080/03081087.2015.1082957

M3 - Article

SN - 0308-1087

VL - 64

SP - 1220

EP - 1235

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

IS - 6

ER -