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On the shape of the numerical range of some tridiagonal matrices

  • Arobinda Ghosh
  • Riddhick Birbonshi
  • Sarita Ojha
  • Kallol Paul
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Abstract

In this paper, we study the elliptical shape of the numerical range of tridiagonal matrices of orders $3,4$ and $5$ with zero entries on the main diagonal, a positive constant value a along the super-diagonal, and sub-diagonal entries are r, sr2, r3, sr4,…, rn−1 & if $n$ is even, and r, sr2, r3, sr4,…, srn−1 & if $n$ is odd, where n denotes the order of the matrix, s>0 and r is a real number. We further investigate the flat portions on the boundary of the numerical range of these n× n matrices in the special case r=−1. Some results given in \cite{chien2011numerical} are obtained as a particular case of our results.

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Cite

BibTeXRIS
BibTeX
@misc{ghosh2026shape,
  title = {{On the shape of the numerical range of some tridiagonal matrices}},
  author = {Arobinda Ghosh and Riddhick Birbonshi and Sarita Ojha and Kallol Paul},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31231},
  eprint = {2609.31231},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31231},
}
RIS
TY  - UNPB
TI  - On the shape of the numerical range of some tridiagonal matrices
AU  - Arobinda Ghosh
AU  - Riddhick Birbonshi
AU  - Sarita Ojha
AU  - Kallol Paul
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31231
UR  - https://arxiv.org/abs/2609.31231
AB  - In this paper, we study the elliptical shape of the numerical range of tridiagonal matrices of orders $3,4$ and $5$ with zero entries on the main diagonal, a positive constant value a along the super-diagonal, and sub-diagonal entries are r, sr^2, r^3, sr^4,…, r^n−1 & if $n$ is even, and r, sr^2, r^3, sr^4,…, sr^n−1 & if $n$ is odd, where n denotes the order of the matrix, s>0 and r is a real number. We further investigate the flat portions on the boundary of the numerical range of these n× n matrices in the special case r=−1. Some results given in \cite{chien2011numerical} are obtained as a particular case of our results.
ER  -