Norms on Complex Matrices Induced by Random Vectors

TL;DR

This paper introduces a new family of matrix norms based on probabilistic methods, extending classical positivity theorems and involving advanced combinatorial and algebraic techniques.

Contribution

It presents a novel class of norms on complex matrices derived from probabilistic and combinatorial frameworks, generalizing Hunter's positivity theorem.

Findings

Defined new probabilistic matrix norms

Generalized Hunter's positivity theorem

Connected norms with trace polynomials and combinatorics

Abstract

We introduce a family of norms on the $n \times n$ complex matrices. These norms arise from a probabilistic framework, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. As a consequence, we obtain a generalization of Hunter's positivity theorem for the complete homogeneous symmetric polynomials.