Reciprocal Lie-Trotter formula

Koenraad Audenaert, F. Hiai

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Pages (from-to)1220-1235
Number of pages16
JournalLinear and Multilinear Algebra
Issue number6
Early online date7 Sept 2015
Publication statusPublished - 2016

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