Reciprocal Lie–Trotter formula

Abstract

Let and be positive semidefinite matrices. The limit of the expression as tends to is given by the well-known Lie–Trotter formula. A similar formula holds for the limit of as tends to , where is the geometric mean of and . In this paper we study the limit of and as tends to instead of , with the ultimate goal of finding an explicit formula, which we call the reciprocal Lie–Trotter formula. We show that the limit of exists and find an explicit formula in a special case. The limit of is shown for matrices only.