Journal
LINEAR & MULTILINEAR ALGEBRA
Volume 64, Issue 6, Pages 1220-1235Publisher
TAYLOR & FRANCIS LTD
DOI: 10.1080/03081087.2015.1082957
Keywords
Primary: 15A42; 15A16; 47A64; antisymmetric tensor power; Grassmannian manifold; Lie-Trotter formula; reciprocal Lie-Trotter formula; log-majorization; operator mean; positive semidefinite matrix; geometric mean
Categories
Funding
- [(C)26400103]
- Grants-in-Aid for Scientific Research [26400103] Funding Source: KAKEN
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Let [GRAPHICS] and [GRAPHICS] be positive semidefinite matrices. The limit of the expression [GRAPHICS] as [GRAPHICS] tends to [GRAPHICS] is given by the well-known Lie-Trotter formula. A similar formula holds for the limit of [GRAPHICS] as [GRAPHICS] tends to [GRAPHICS] , where [GRAPHICS] is the geometric mean of [GRAPHICS] and [GRAPHICS] . In this paper we study the limit of [GRAPHICS] and [GRAPHICS] as [GRAPHICS] tends to [GRAPHICS] instead of [GRAPHICS] , with the ultimate goal of finding an explicit formula, which we call the reciprocal Lie-Trotter formula. We show that the limit of [GRAPHICS] exists and find an explicit formula in a special case. The limit of [GRAPHICS] is shown for [GRAPHICS] matrices only.
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