Dual Hybrid Numbers and Their Hybrid Matrix Representations

Abstract

Hybrid numbers are introduced as a linear combination of complex, dual and hyperbolic numbers, where hybrid units satisfy \(i^{2}=-1\), \(\epsilon ^{2}=0\) and \(h^{2}=1\), respectively. The main motivation of this study is to present dual hybrid numbers and dual hybrid matrices. In this context, we first give some results of hybrid numbers related to complex and hyperbolic numbers. Afterward, we define hybrid matrix representation of a dual hybrid matrix by introducing dual hybrid matrices whose elements consist of dual hybrid numbers. Since dual hybrid numbers are not commutative, it is necessary to examine the right and left eigenvalues of dual hybrid matrices separately. In this article, we focus on the right eigenvalues of dual hybrid matrices and give some of their properties. Finally, we provide hybrid matrix representations of dual hybrid numbers and support our results with an example