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A simplification of the C-realizability criterion for the Nonnegative Inverse Eigenvalue Problem for integers

Alberto Borobia, Roberto Canogar

Abstract

A multiset $Λ=\{λ_1,\ldots,λ_n\}$ of complex numbers is said to be realizable whenever there exists a nonnegative matrix of order $n$ with spectrum $Λ$. One of the broadest criterion that guarantees realizability is the $C-$realizability. It says that $Λ$, with real numbers, is $C-$realizable if it can be obtained starting from $n$ basic multisets $\{0\},\ldots,\{0\}$ by successively applying any finite number of times any of the following rules: (a) join two of the multisets; (b) increase by $ε>0$ the Perron root of one of the multisets; (c) increase by $ε>0$ the Perron root of one of the multisets and simultaneously increase or decrease by $ε$ any other value of the same multiset.

A simplification of the C-realizability criterion for the Nonnegative Inverse Eigenvalue Problem for integers

Abstract

A multiset of complex numbers is said to be realizable whenever there exists a nonnegative matrix of order with spectrum . One of the broadest criterion that guarantees realizability is the realizability. It says that , with real numbers, is realizable if it can be obtained starting from basic multisets by successively applying any finite number of times any of the following rules: (a) join two of the multisets; (b) increase by the Perron root of one of the multisets; (c) increase by the Perron root of one of the multisets and simultaneously increase or decrease by any other value of the same multiset.
Paper Structure (14 sections, 23 theorems, 28 equations, 2 figures)

This paper contains 14 sections, 23 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. An alternative approach to $C-$realizability
  3. $C-$realizability for integers
  4. States and types of moves for states
  5. Laws for moves
  6. $C^*_{\mathbb{Z},6}$ and $C^*_{\mathbb{Z},3}-$realizability for states
  7. Swapping moves
  8. Swapping a valid move of type 2
  9. Swapping an invalid move of type 2
  10. A technical lemma
  11. Main Theorem
  12. Final remarks
  13. Extension of the main theorem to rationals and to reals
  14. Reordering the moves

Key Result

Theorem 1

Let $\Lambda$ and $\Gamma$ be two realizable multisets of $\mathbb{C}$, then $\Lambda\cup \Gamma$ is realizable.

Figures (2)

  • Figure 1: (left) ordering the entries on each multiset, (right) without ordering the entries on each multiset
  • Figure 2: (left) tangled tree, (right) untangled tree.

Theorems & Definitions (42)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Definition 8
  • Definition 9
  • Theorem 10
  • ...and 32 more