On (conditional) positive semidefiniteness in a matrix-valued context

TL;DR

This paper extends Schoenberg's theorem from scalar to matrix-valued functions, exploring conditions for positive semidefiniteness and positivity preservation of matrix exponentials in a multivariate setting.

Contribution

It generalizes classical scalar results to matrix-valued functions and investigates the limitations of positivity preservation in this broader context.

Findings

Extension of Schoenberg's theorem to matrix-valued functions

Identification of conditions for positive semidefinite exponentials

Analysis of the failure of positivity preservation in matrix case

Abstract

In a nutshell, we intend to extend Schoenberg's classical theorem connecting conditionally positive semidefinite functions $F\colon \mathbb{R}^n \to \mathbb{C}$, $n \in \mathbb{N}$, and their positive semidefinite exponentials $\exp(tF)$, $t > 0$, to the case of matrix-valued functions $F \colon \mathbb{R}^n \to \mathbb{C}^{m \times m}$, $m \in \mathbb{N}$. Moreover, we study the closely associated property that $\exp(t F(- i \nabla))$, $t>0$, is positivity preserving and its failure to extend directly in the matrix-valued context.