Total-positivity preservers

TL;DR

This paper characterizes all entrywise transformations that preserve total positivity or non-negativity in matrices, showing they must be constant or linear, with extensions to symmetric matrices and specific completion problems.

Contribution

It provides a complete classification of total-positivity preservers for fixed-dimension matrices, including symmetric cases and related completion problems.

Findings

Only constant or linear transforms preserve total positivity/non-negativity.

Classification extends to symmetric matrices.

Proofs involve solving totally positive completion problems.

Abstract

We prove that the only entrywise transforms of rectangular matrices which preserve total positivity or total non-negativity are either constant or linear. This follows from an extended classification of preservers of these two properties for matrices of fixed dimension. We also prove that the same assertions hold upon working only with symmetric matrices; for total-positivity preservers our proofs proceed through solving two totally positive completion problems.