Extremal Schur Multipliers

TL;DR

This paper investigates extremal elements in the set of matrices with bounded Schur multiplier norm, establishing rank bounds and characterizations for positive matrices with unit diagonal.

Contribution

It provides a new inequality relating the rank of extremal matrices to their dimensions and characterizes extremal elements for positive matrices with unit diagonal.

Findings

Extremal matrices have rank at most (m+n)^(1/2).

Positive matrices with unit diagonal have extremal elements with rank at most n^(1/2).

Characterizations of extremal elements are provided for specific matrix classes.

Abstract

The Schur product of two complex m x n matrices is their entry wise product. We show that an extremal element X in the convex set of m x n complex matrices of Schur multiplier norm at most 1 satisfies the inequality rank(X) =< (m +n)^(1/2) . For positive n x n matrices with unit diagonal, we give a characterization of the extremal elements, and show that such a matrix satisfies rank(X) =< n^(1/2).