It is known that the hybrid numbers are generalizations of complex, hyperbolic and dual numbers. Recently, they have attracted the attention of many scientists. At this paper, we provide the Euler’s and De Moivre’s formulas for the 4×4 matrices associated with hybrid numbers by using trigonometric identities. Also, we give the roots of the matrices of hybrid numbers. Moreover, we give some illustrative examples to support the main formulas.
In this paper, we obtain Euler’s and De Moivre’s formulas for the [Formula: see text] matrix representation of Pauli quaternions. Moreover, we provide De Moivre’s formula for the light-like Pauli quaternions. Additionally, we give the [Formula: see text] roots of the matrix representation of Pauli quaternions. Moreover, we exemplify some of the results with illustrative examples to support the main formulas.
In the literature, Holditch theorem was obtained under periodic rotation and translation motions in [H. Holditch, Geometrical theorem, Q. J. Pure Appl. Math. 2 (1858) 38] or periodic shear and translation motions in [O. Röschel, Der satz von Holditch in der isotropen ebene. Abh. Braunschweig. Wiss. Ges. 36 (1984) 27–32]. In this paper, by introducing the projection of a vector onto a plane, scalar area, area vector of a surface, we investigate Holditch theorem under periodic rotation, translation and shear motions. We give two interpretations for the Holditch type theorem in Galilean space.
The Pell numbers, named after the English diplomat and mathematician John Pell, are studied by many authors. At this work, by inspiring the definition harmonic numbers, we define harmonic Pell numbers. Moreover, we construct one type of symmetric matrix family whose elements are harmonic Pell numbers and its Hadamard exponential matrix. We investigate some linear algebraic properties and obtain inequalities by using matrix norms. Furthermore, some summation identities for harmonic Pell numbers are obtained. Finally, we give a MATLAB-R2016a code which writes the matrix with harmonic Pell entries and calculates some norms and bounds for the Hadamard exponential matrix.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.