Localization theorems for matrices and bounds for the zeros of polynomials over a quaternion division algebra

TL;DR

This paper extends classical eigenvalue localization theorems and stability criteria to quaternionic matrices, providing bounds for polynomial zeros over quaternions, thus broadening spectral analysis tools in quaternion algebra.

Contribution

It introduces quaternionic versions of Ostrowski, Brauer, and Gerschgorin theorems, and offers a stability condition and zero bounds for quaternionic polynomials, advancing spectral theory in quaternion matrices.

Findings

Derived Ostrowski and Brauer type theorems for quaternionic eigenvalues

Generalized Gerschgorin theorems for quaternionic matrices

Provided bounds for zeros of quaternionic polynomials

Abstract

In this paper, Ostrowski and Brauer type theorems are derived for the left and right eigenvalues of a quaternionic matrix. Generalizations of Gerschgorin type theorems are discussed for the left and the right eigenvalues of a quaternionic matrix. Thereafter a sufficient condition for the stability of a quaternionic matrix is given that generalizes the stability condition for a complex matrix. Finally, a characterization of bounds for the zeros of quaternionic polynomials is presented.