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

TL;DR

This paper demonstrates that for Hermitian matrices A and B, the matrix exponential e^{At+B} can be expressed as a bilateral Laplace transform of a matrix measure supported on the spectrum's convex hull, extending the classical scalar case.

Contribution

It establishes a representation of e^{At+B} as a Laplace transform of a matrix measure for Hermitian matrices, detailing properties of the measure's support and Hermitian nature.

Findings

e^{At+B} equals the bilateral Laplace transform of a matrix measure.

The measure's support lies within the convex hull of A's spectrum.

If B is Hermitian, the measure's values are Hermitian matrices.

Abstract

Given a pair $A,B$ of matrices of size $n\times n$, we consider the matrix function $e^{At+B}$ of the variable $t\in\mathbb{C}$. If the matrix $A$ is Hermitian, the matrix function $e^{At+B}$ is representable as the bilateral Laplace transform of a matrix-valued measure $M(d\lambda)$ compactly supported on the real axis: $$e^{At+B}=\int{}e^{\lambda t}\,M(d\lambda).$$ 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.