Monotonicity for entrywise functions of matrices

TL;DR

This paper characterizes functions that preserve positivity, monotonicity, and convexity when applied entrywise to matrices, with examples including fractional powers and related majorization properties.

Contribution

It provides a comprehensive characterization of functions maintaining positive semidefinite order properties when applied entrywise to matrices.

Findings

Fractional power functions preserve positive semidefiniteness.

Weak majorizations related to entrywise functions are established.

Characterization results apply to functions on specific intervals.

Abstract

We characterize real functions $f$ on an interval $(-\alpha,\alpha)$ for which the entrywise matrix function $[a_{ij}] \mapsto [f(a_{ij})]$ is positive, monotone and convex, respectively, in the positive semidefiniteness order. Fractional power functions are exemplified and related weak majorizations are shown.