Quaternion-valued exponential matrices and its fundamental properties

Abstract

Quaternion differential equations (QDEs) are a new kind of differential equations which differ from ordinary differential equations. Our aim is to get the exponential matrices for the QDE which is useful for finding the solution of quaternion-valued differential equations, also, we know that linear algebra is very useful to calculate the exponential for a matrix but the solution of QDE is not a linear space. Due to the noncommutativity of the quaternion, the solution set of QDE is a right free module. For this, we must read some basic concepts on Quaternions such as eigenvalues, eigenvectors, Wronskian and the difference between quaternion and complex eigenvalues and eigenvectors; by using the right eigenvalue method for quaternions we developed a fundamental matrix which is useful to construct the exponential matrices which perform a great role in solving the QDEs.