#### Date of Award

9-1-2021

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Kocik, Jerzy

#### Abstract

There are four normed division algebras over real numbers, namely real numbers, complex numbers, quaternions, and octonions. Lack of commutativity and associativity make it difficult to investigate algebraic and geometric properties of octonions. Eigenvalue problem of octonionic Hermitian matrices is one of the interesting studies where we can see this difficulty of extending the basic properties from complex Hermitian matrices to octonionic Hermitian matrices. This includes the notion of orthogonality and decomposition of a Hermitian matrix using its eigenvalues and eigenvectors.Liping Huang and Wasin So derived explicit formulas for computing the roots of quaternionic quadratic equations. We extend their work to octonionic case and solve octonionic left quadratic equations. We represent left spectrum of two by two octonionic Hermitian matrix using the solutions to corresponding octonionic left quadratic equation and identify the family of matrices which admit non-real left eigenvalues. For three by three case we review previous work by Tevian Dray and Corinne Manogue of real eigenvalue problem and study characteristic equations which admit non-real roots that are correspond to non-real left eigenvalues. Finally, we discuss the right spectrum using the associator method, and provide examples using "pyoctonion" python library. Interesting applications and open problems for future studies in this literature are also included.

#### Access

This dissertation is Open Access and may be downloaded by anyone.