The Matrix Function $$\varvec{e^{tA+B}}$$ e t A + B is Representable as the Laplace Transform of a Matrix Measure

Abstract

Given a pair \(A,B\) of matrices of size \(n\times n\), we consider the matrix function \(e^{tA+B}\) of the variable \(t\in \mathbb {C}\). If the matrix \(A\) is Hermitian, the matrix function \(e^{tA+B}\) is representable as the bilateral Laplace transform of a matrix-valued measure \(M(d\lambda )\) compactly supported on the real axis: $$\begin{aligned} e^{tA+B}=\int {}e^{t\lambda }\,M(d\lambda ). \end{aligned}$$ The values of the measure \(M(d\lambda )\) are matrices of size \(n\times n\), 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 \(M(d\lambda )\) are Hermitian matrices. The measure \(M(d\lambda )\) is not necessarily non-negative.