Localization of the Spectrum of Matrices with Quaternion Entries

Grant Winner

Dean

Abstract

As the complex numbers are extensions of the real numbers, the quaternions are extensions of the complex numbers. Quaternions are not only part of contemporary mathematics but also widely used in 3D computer graphics, control theory, signal processing, altitude control, and quantum mechanics. A better understanding of quaternions and matrices of quaternions is important for further application of quaternion theory to state-of-the-art science and technology.

Matrix theory (over the real and complex number fields) is fundamental to mathematics, while eigenvalues of matrices are a central topic of matrix theory. The main obstacle in the study of quaternion matrices is the noncommutative multiplication of quaternions. The spectrum of a matrix is the collection of all eigenvalues of the matrix. The basic theory on the right eigenvalues (spectrum) of quaternion matrices has been well established; however, little is known for the left eigenvalues. It has been evident that the right eigenvalues (up to equivalence classes) of quaternion matrices resemble the eigenvalues of complex matrices; while the left eigenvalues "behave" very differently.

This project addresses several topics about the spectrum (spectra) of matrices with quaternion entries, among them are the distribution of the left eigenvalues and higher order Ger\v{s}gorin regions for the right eigenvalues. In particular, we will determine whether our conjecture is true that most of the left eigenvalues (in case of infinitely many) are located on the surface of so-called q-balls. The main tool for the research is the method of function continuity and matrix complex conjugation. The results should lead to a deeper understanding of the matrix theory over division algebra and the usefulness of quaternion matrices in modern technology.