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Location of the zeros of quaternionic polynomials using matrix tools

N. A. Rather, Naseer Ahmad Wani, Ishfaq Dar

Abstract

Using a variety of matrix techniques, the problem of locating the left eigenvalues of the quaternion companion matrices are investigated in this paper. In a recent paper, Dar et al. [6], proved that the zeros of a quaternionic polynomial and the left eigenvalues of corresponding companion matrix are same. In view of this, we use various newly developed matrix techniques to prove various results concerning the location of the zeros of regular polynomials of a quaternionic variable with quaternionic coefficients, which include an extension of the result of A. L. Cauchy as well.

Location of the zeros of quaternionic polynomials using matrix tools

Abstract

Using a variety of matrix techniques, the problem of locating the left eigenvalues of the quaternion companion matrices are investigated in this paper. In a recent paper, Dar et al. [6], proved that the zeros of a quaternionic polynomial and the left eigenvalues of corresponding companion matrix are same. In view of this, we use various newly developed matrix techniques to prove various results concerning the location of the zeros of regular polynomials of a quaternionic variable with quaternionic coefficients, which include an extension of the result of A. L. Cauchy as well.
Paper Structure (5 sections, 14 theorems, 49 equations)

This paper contains 5 sections, 14 theorems, 49 equations.

Table of Contents

  1. INTRODUCTION
  2. Preliminary
  3. Main Results
  4. Auxiliary Results
  5. Proof of the main theorems

Key Result

Theorem A

All the eigenvalues of a $n \times n$ complex matrix $A=(a_{\mu\nu})$ are contained in the union of n Geršgorin discs defined by $D_{\mu}=\{ z \in \mathbb{C}:|z-a_{\mu\mu}|\leq \sum_{{\nu=1}\atop{\nu\neq\mu}}^n |a_{\mu\nu}|\}$.

Theorems & Definitions (16)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem 1
  • Remark 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • ...and 6 more