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Polynomials that preserve nonnegative monomial matrices

Benjamin J. Clark, Pietro Paparella

TL;DR

This work addresses which polynomials preserve entrywise nonnegativity on nonnegative matrices, focusing on the restricted class of nonnegative monomial matrices. It develops a concrete polynomial calculus for monomial matrices by exploiting the Frobenius normal form and the decomposition into $K_x$ blocks, yielding a closed-form for $p(K_x)$ in terms of the $rmod n$ parts $p_{(r,n)}$. The central result proves that the Clark–Paparella necessary condition for $\\mathscr{P}_n$ is also sufficient for $\\mathscr{P}_n^{\\rm mon}$, characterizing $p$ via the requirement that all $p_{(r,n)} \\in \\mathscr{P}_1$. This advances understanding of nonnegative matrix polynomials and informs the nonnegative inverse eigenvalue problem by providing explicit, block-wise positivity criteria and computations.

Abstract

A recently-established necessary condition for polynomials that preserve the class of entrywise nonnegative matrices of a fixed order is shown to be necessary and sufficient for the class of nonnegative monomial matrices. Along the way, we provide a formula for computing an arbitrary power of a monomial matrix and a formula for computing the polynomial of a nonnegative monomial matrix.

Polynomials that preserve nonnegative monomial matrices

TL;DR

This work addresses which polynomials preserve entrywise nonnegativity on nonnegative matrices, focusing on the restricted class of nonnegative monomial matrices. It develops a concrete polynomial calculus for monomial matrices by exploiting the Frobenius normal form and the decomposition into blocks, yielding a closed-form for in terms of the parts . The central result proves that the Clark–Paparella necessary condition for is also sufficient for , characterizing via the requirement that all . This advances understanding of nonnegative matrix polynomials and informs the nonnegative inverse eigenvalue problem by providing explicit, block-wise positivity criteria and computations.

Abstract

A recently-established necessary condition for polynomials that preserve the class of entrywise nonnegative matrices of a fixed order is shown to be necessary and sufficient for the class of nonnegative monomial matrices. Along the way, we provide a formula for computing an arbitrary power of a monomial matrix and a formula for computing the polynomial of a nonnegative monomial matrix.
Paper Structure (4 sections, 5 theorems, 31 equations)

This paper contains 4 sections, 5 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminary results
  4. Main results

Key Result

Lemma 3.1

If $A \in \mathsf{GP}_n$, then there is a permutation matrix $Q$ such that where $1 \leq k \leq n$, $y = Q^\top x$, and $y_i \in \mathbb{C}^{n_i}$ is the partition of $y$ into $k$ blocks of size $n_i$ for $i \in \langle{k}\rangle$.

Theorems & Definitions (14)

  • Definition 2.1: cp2022
  • Remark 2.2
  • Lemma 3.1: Frobenius normal form for monomial matrices
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • ...and 4 more