The matrix function e^tA+B is representable as the Laplace transform of a matrix measure

2016/09/13 by Katsnelson, Victor
#15A16 #44A10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1609.03870

Abstract

Given a pair A,B of matrices of size n× n, we consider the matrix function e^At+B of the variable t∈ℂ. If the matrix A is Hermitian, the matrix function e^At+B is representable as the bilateral Laplace transform of a matrix-valued measure M(dλ) compactly supported on the real axis: e^At+B=∫e^λt M(dλ). The values of the measure M(dλ) are matrices of size n× n, the support of this measure is contained in the convex hull of the spectrum of A. If the matrix B is also Hermitian, then the values of the measure M(dλ) are Hermitian matrices. The measure M(dλ) is not necessarily non-negative.

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