Local characterizations for the matrix monotonicity and convexity of fixed order

TL;DR

This paper introduces local characterizations for matrix monotonicity and convexity of fixed order using integral representations that relate Loewner and Kraus matrices to Hankel matrices, simplifying existing methods.

Contribution

It provides new local characterizations for matrix convexity and monotonicity, extending Kraus's original work and clarifying their relationship.

Findings

New integral representations connect Loewner and Kraus matrices to Hankel matrices.

Simplified approach to characterizing matrix monotonicity.

Extended characterization of matrix convexity by Kraus.

Abstract

We establish local characterizations of matrix monotonicity and convexity of fixed order by giving integral representations connecting the Loewner and Kraus matrices, previously known to characterize these properties, to respective Hankel matrices. Our results are new already in the general case of matrix convexity and our approach significantly simplifies the corresponding work on matrix monotonicity. We also obtain an extension of the original characterization for matrix convexity by Kraus, and tighten the relationship between monotonicity and convexity.