Solve the linear quaternion-valued differential equations having multiple eigenvalues

TL;DR

This paper addresses the challenge of constructing fundamental matrices for linear quaternion-valued differential equations with multiple eigenvalues, introducing new methods to find missing solutions and a novel determinant definition for quaternion matrices.

Contribution

It introduces a new approach to handle multiple eigenvalues in quaternion differential equations and proposes a permutation-based determinant for quaternion matrices.

Findings

New method to construct fundamental matrices with multiple eigenvalues

Introduction of a permutation-based quaternion determinant

Enhanced analysis of quaternion differential equations

Abstract

The theory of two-dimensional linear quaternion-valued differential equations (QDEs) was recently established (see Kou and Xia, SAPM). Some profound differences between QDEs and ODEs were observed. Also, an algorithm to evaluate the fundamental matrix by employing the eigenvalues and eigenvectors was presented in [Kou and Xia, SAPM]. However, the fundamental matrix can be constructed providing that the eigenvalues are simple. If the linear system has multiple eigenvalues, how to construct the fundamental matrix? In particular, if the number of independent eigenvectors might be less than the dimension of the system. That is, the numbers of the eigenvectors is not enough to construct a fundamental matrix. How to find the "missing solutions"? The main purpose of this paper is to answer this question. Furthermore, Caley determinant for Quaternion-valued matrix was adopted to proceed the theory of QDEs in [Kou and Xia, SAPM]. One big disadvantage of Caley determinant is that it can be expanded along the different rows and columns. This may lead to different results due to non-commutativity of the quaternions. This approach is not convenient to be used. The novel definition of determinant for Quaternion-valued matrix based on permutation is introduced to analyze the theory. This newly definition of determinant has great advantage compare to Calay determinant.