A New Method for Examination of Eigenvalue-Eigenvector Theory for Hyperdual Number Matrices

Abstract

Hyperdual number matrices have important applications in numerical differentiation and multibody kinematics. In this paper, we introduce concepts of determinant, characteristic polynomial, eigenvalues, and eigenvectors over hyperdual number matrices. First, we review these concepts for dual number matrices, then introduce a new approach to the determinant of hyperdual number matrices inspired by dual number matrix combination. We define the characteristic polynomial of hyperdual number matrices. Then, we demonstrate the necessary and sufficient conditions under which the characteristic roots of a hyperdual number matrix can be considered eigenvalues. Based on these, we observe that hyperdual number matrices may have at most eigenvalues, no eigenvalues at all, or infinitely many eigenvalues, as do dual number matrices. Hence, we determine the eigenvectors of a hyperdual number matrix. We provide some examples to illustrate our theorems and results. As a final step, a discussion of these results is outlined in tridiagonal matrices for the dual and hyperdual cases with examples.