Similarity and consimilarity of hyper-dual generalized quaternions

Abstract

The aim of this paper is to investigate similarity and consimilarity of hyper-dual generalized quaternions and their matrices. For this purpose, we give different conjugates according to the generalized quaternionic units $i,j,k$. We present $i$-consimilarity of hyper-dual generalized quaternions and their matrices except hyper-dual $\frac{1}{4}$-quaternions. For the generalization consisting of hyper-dual coefficients quaternion and split quaternion, we search $j$-consimilarity and $k$-consimilarity with the help of $j$-conjugate and $k$-conjugate. We also give $i,j,k$-coneigenvalues and $i,j,k$-coneigenvectors of the matrices of these generalizations. In addition, we examine right coneigenvalue problem in generalized quaternion matrices for real and split quaternions. The complex matrix representation obtained through the complex adjoint matrix representation of this generalization is introduced, and its properties are presented. Besides, we give algebraic methods for the concept of right coneigenvalues and coneigenvectors for matrices, which are the generalization of real quaternion and split quaternion.