Circularity and Polygonal Structure of Numerical Ranges for Non-Negative Matrices

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@article{1706122,
      author = {Joel S. Barasa , Shem Away, Lucy Walingo Chikamai},
      title = {Circularity and Polygonal Structure of Numerical Ranges for Non-Negative Matrices},
      journal = {Iconic Research And Engineering Journals},
      year = {2024},
      volume = {8},
      number = {2},
      pages = {88-93},
      issn = {2456-8880},
      url = {https://www.irejournals.com/formatedpaper/1706122.pdf},
      abstract = {This study investigates the circularity and polygonal structure of numerical ranges for non-negative matrices. We provide a comprehensive characterization of non-negative matrices with circular numerical ranges and derive conditions for polygonal numerical ranges. Our main results show that a non-negative matrix has a circular numerical range centered at the origin if and only if it is unitarily equivalent to a scalar multiple of a doubly stochastic matrix. Furthermore, we prove that the numerical range of a non-negative matrix is a regular polygon with k vertices if and only if the matrix is unitarily equivalent to a direct sum of k cyclic permutation matrices. We explore the relationships between matrix structure (irreducibility, sparsity, and symmetry) and numerical range geometry, supported by computational methods for visualization and analysis. The study establishes connections between numerical range geometry and applications in Markov chains, quantum information theory, and graph theory. Our findings extend existing theory, provide new matrix characterizations based on numerical range geometry, and offer computational tools for further research. This work contributes to a deeper understanding of non-negative matrices and their properties, with potential implications for various fields in mathematics, physics, and computer science.},
      keywords = {Numerical Range, Non-Negative Matrices, Circularity, Polygonal Structure, Matrix Analysis, Operator Theory},
      month = {August},
  }