Melek Erdoğdu

Bio: Melek Erdoğdu is an academic researcher. The author has contributed to research in topics: Minkowski space & Mathematical analysis. The author has an hindex of 7, co-authored 15 publications receiving 166 citations.

On Eigenvalues of Split Quaternion Matrices

Abstract: The main purpose of this paper is to set a method of finding eigenvalues of split quaternion matrices. In particular, we will give an extension of Gershgorin theorem, which is one of the fundamental theorems of complex matrix theory, for split quaternion matrices.

Geometry of Hasimoto Surfaces in Minkowski 3-Space

Abstract: In this paper, we investigate the Hasimoto surfaces in Minkowski 3-space. We discussed the geometric properties of Hasimoto surfaces in $\mathbb {M}^{3}$ for three cases. The Gaussian and mean curvature of Hasimoto surface are found for each case. Then, we give the characterization of parameter curves of Hasimoto surfaces in $\mathbb {M}^{3}.$

On Complex Split Quaternion Matrices

Abstract: In this paper, we present some important properties of complex split quaternions and their matrices. We also prove that any complex split quaternion has a 4 × 4 complex matrix representation. On the other hand, we give answers to the following two basic questions “If AB = I, is it true that BA = I for complex split quaternion matrices?” and “How can the inverse of a complex split quaternion matrix be found?”. Finally, we give an explicit formula for the inverse of a complex split quaternion matrix by using complex matrices.

On the eigenvalues and eigenvectors of a lorentzian rotation matrix by using split quaternions

Abstract: In this paper, we examine eigenvalue problem of a rotation matrix in Minkowski 3 space by using split quaternions. We express the eigenvalues and the eigenvectors of a rotation matrix in term of the coefficients of the corresponding unit timelike split quaternion. We give the characterizations of eigenvalues (complex or real) of a rotation matrix in Minkowski 3 space according to only first component of the corresponding quaternion. Moreover, we find that the casual characters of rotation axis depend only on first component of the corresponding quaternion. Finally, we give the way to generate an orthogonal basis for \({\mathbb{E}^{3}_{1}}\) by using eigenvectors of a rotation matrix.

Introduction to Hybrid Numbers

Abstract: In this study, we define a new non-commutative number system called hybrid numbers. This number system can be accepted as a generalization of the complex $$\left( {\mathbf {i}}^{2}=-1\right) $$ , hyperbolic $$\left( {\mathbf {h}} ^{2}=1\right) $$ and dual number $$\left( \varvec{\varepsilon }^{2}=0\right) $$ systems. A hybrid number is a number created with any combination of the complex, hyperbolic and dual numbers satisfying the relation $$\mathbf { ih=-hi=i}+\varvec{\varepsilon }.$$ Because these numbers are a composition of dual, complex and hyperbolic numbers, we think that it would be better to call them hybrid numbers instead of the generalized complex numbers. In this paper, we give some algebraic and geometric properties of this number set with some classifications. In addition, we examined the roots of a hybrid number according to its type and character.

On Complex Split Quaternion Matrices

Abstract: In this paper, we present some important properties of complex split quaternions and their matrices. We also prove that any complex split quaternion has a 4 × 4 complex matrix representation. On the other hand, we give answers to the following two basic questions “If AB = I, is it true that BA = I for complex split quaternion matrices?” and “How can the inverse of a complex split quaternion matrix be found?”. Finally, we give an explicit formula for the inverse of a complex split quaternion matrix by using complex matrices.

On Hermitian Solutions of the Split Quaternion Matrix Equation $$AXB+CXD=E$$ A X B + C X D = E

Abstract: In this paper, we discuss Hermitian solutions of split quaternion matrix equation $$AXB+CXD=E,$$ where X is an unknown split quaternion Hermitian matrix, and A, B, C, D, E are known split quaternion matrices with suitable size. The objective of this paper is to establish a necessary and sufficient condition for the existence of a solution and a solution formulas. Moreover, we provide numerical algorithms and numerical examples to exemplify the results.