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Non-negative persymmetric realizability of certain classes of spectra

Nayanthara, Noufal Asharaf

Abstract

Identifying the collection of scalars that represent a non-negative matrix's eigenvalues is known as the non-negative inverse eigenvalue problem (NIEP). Conditions for the existence of a non-negative matrix with a certain spectrum are examined in this work. The classical NIEP restricted to non-negative matrices having a persymmetric structure is the persymmetric non-negative inverse eigenvalue problem (PNIEP). We resolve the open problem stated in \cite{Ana}, and furthermore, equivalence of the PNIEP and NIEP is established for trace-zero spectra of five complex numbers. Also we obtain new sufficient conditions for the realizability of certain classes of spectra with non-negative persymmetric matrix realization. Based on generic structural features of persymmetric matrices and their characteristic polynomials, our method is constructive in nature. The effects of perturbations on imaginary part of complex eigenvalues are also analyzed and some perturbation results are derived.

Non-negative persymmetric realizability of certain classes of spectra

Abstract

Identifying the collection of scalars that represent a non-negative matrix's eigenvalues is known as the non-negative inverse eigenvalue problem (NIEP). Conditions for the existence of a non-negative matrix with a certain spectrum are examined in this work. The classical NIEP restricted to non-negative matrices having a persymmetric structure is the persymmetric non-negative inverse eigenvalue problem (PNIEP). We resolve the open problem stated in \cite{Ana}, and furthermore, equivalence of the PNIEP and NIEP is established for trace-zero spectra of five complex numbers. Also we obtain new sufficient conditions for the realizability of certain classes of spectra with non-negative persymmetric matrix realization. Based on generic structural features of persymmetric matrices and their characteristic polynomials, our method is constructive in nature. The effects of perturbations on imaginary part of complex eigenvalues are also analyzed and some perturbation results are derived.
Paper Structure (10 sections, 16 theorems, 19 equations)

This paper contains 10 sections, 16 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Non-negative Inverse Eigenvalue Problem
  3. Main Results
  4. Sufficient Conditions for Persymmetric realizability
  5. Sufficient Conditions for Persymmetric Realizability of Spectra with Single Positive Eigenvalue
  6. Sufficient Conditions for Persymmetric Realizability of Five-Element Spectra
  7. Realizability with a perturbation of Eigenvalues
  8. A Prescribed List of Four Complex Numbers
  9. A prescribed list of five complex numbers
  10. Conclusion

Key Result

Theorem 2.1

low Let $\Delta:=\{\delta_1, \delta_2, \delta_3\}$be a list of complex numbers with $\delta_1 \geq |\delta_j|, j=2,3.$ Then $\Delta$ is realizable if and only if $\Delta(A) = \Bar {\Delta}(A), s_1 \geq0\,\, \text{and}\,\, s_1^2 \leq 3s_2.$

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.1
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 3.1
  • ...and 29 more