Split Quaternions Research Articles

Hyper-dual split quaternions and rigid body motion

In this paper, we introduce and define hyper-dual split vectors by using hyper-dual numbers. We define three subsets; two of them are subsets of Lorentzian unit hyper-dual sphere and the other is a subset of hyperbolic unit hyper-dual sphere. We show that to each element of these subsets corresponds two intersecting perpendicular lines in Minkowski space. We also introduce and define hyper-dual split quaternions with their basic algebraic properties. We give the geometric interpretation of hyper-dual split quaternions, and we define three new operators in Minkowski space. We show that these operators turn each two intersecting perpendicular lines again to two intersecting perpendicular lines in Minkowski space. Examples are also provided to illustrate our theorems and results.

ON BÄCKLUND TRANSFORMATIONS WITH SPLIT QUATERNIONS

The present paper deals with the introduction of Bäcklund Transformations with split quaternions in Minkowski space. Firstly, we tersely summarized the basic concepts of split quaternion theory and Bishop Frames of non-null curves in Minkowski space. Then, for Bäcklund transformations defined with each case of non-null curves, we give relationships between Bäcklund transformations and split quaternions. It is also presented some special propositions for transformations constructed with split quaternions. At the end, results obtained with the mathematical model have been evaluated.

The Moore-Penrose inverses of split quaternions

ABSTRACT In this paper, we find the roots of lightlike quaternions. By introducing the concept of the Moore-Penrose inverse in split quaternions, we solve the linear equations axb = d, xa = bx, and axb = x. As applications, we obtain necessary and sufficient conditions for two split quaternions to be similar or pseudosimilar.

Real matrix representations of complex split quaternions with applications

In this paper, we introduce 8×8 real matrix representations of complex split quaternions. Then, the relations between real matrix representations of split and complex split quaternions are stated. Moreover, we investigate some linear split and complex split quaternionic equations with split Fibonacci and complex split Fibonacci quaternion coefficients. Finally, we also give some numerical examples as applications of real matrix representation of complex split quaternions.

The Moore–Penrose Inverse and Singular Value Decomposition of Split Quaternions

Using the concept of a transposition anti-involution in Clifford algebra $$C \, \ell _{1,1}$$ and the isomorphisms $${\mathbb {H}}_s \cong C \, \ell _{1,1} \cong \text {Mat}(2,{\mathbb {R}}),$$ where $${\mathbb {H}}_s$$ is the algebra of split quaternions and $$\text {Mat}(2,{\mathbb {R}})$$ is the algebra of $$2 \times 2$$ real matrices, one can find the Moore–Penrose inverse $$q^{+}$$ of a non-zero non-invertible split quaternion q. In particular, using a well-known algorithm for finding the Moore-Penrose inverse $$Q^{+}$$ of a non-zero $$2 \times 2$$ matrix Q of rank 1, one can give four governing equations that the (unique) split quaternion $$q^{+}$$ corresponding to $$Q^{+}$$ must satisfy. We show how a dyadic expansion and a Singular Value Decomposition can be found for any split quaternion q, and we relate them to $$q^{+}$$ . Results presented in this paper may be useful in a plethora of recent applications of split quaternions.

Cramer’s rule over quaternions and split quaternions: A unified algebraic approach in quaternionic and split quaternionic mechanics

This paper aims to present, in a unified manner, Cramer’s rule which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, studies Cramer’s rule for the system of v-quaternionic linear equations by means of a complex matrix representation of v-quaternion matrices, and gives an algebraic technique for solving the system of v-quaternionic linear equations. This paper also gives a unification of algebraic techniques for Cramer’s rule in quaternionic and split quaternionic mechanics.

Split Quaternion-Valued Twin-Multistate Hopfield Neural Networks

Complex-valued Hopfield neural networks have been extended to 4-dimensional models using quaternions and commutative quaternions. The algebra of split quaternions is another 4-dimensional hypercomplex number system. In this work, a split quaternion-valued Hopfield neural network is defined. By the computer simulations using the twin-multistate activation function and projection rule, we evaluate the noise tolerance.

A general setting for functions of Fueter variables: differentiability, rational functions, Fock module and related topics

We develop some aspects of the theory of hyperholomorphic functions whose values are taken in a Banach algebra over a field—assumed to be the real or the complex numbers—and which contains the field. Notably, we consider Fueter expansions, Gleason’s problem, the theory of hyperholomorphic rational functions, modules of Fueter series, and related problems. Such a framework includes many familiar algebras as particular cases. The quaternions, the split quaternions, the Clifford algebras, the ternary algebra, and the Grassmann algebra are a few examples of them.

Quadratic Split Quaternion Polynomials: Factorization and Geometry

We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split quaternions.

On the split quaternion matrix equation $$AX=B$$

In the paper, we consider the split quaternion matrix equation $$AX=B$$. We design several real representations of split quaternion matrix to transform the above split quaternion matrix equation into some real matrix equations. By using this method, we give some necessary and sufficient conditions for $$AX=B$$ to have a X or $$X=\pm X^{\star }$$ solution and derive the expressions of solutions when equation is solvable, where $$X^{\star } \in \{X^*, X^\eta , X^{\eta *}\}$$, $$X^*$$ is the conjugate transpose of X, for $$\eta \in \{i, j, k\}$$, $$ X^{\eta }, X^{\eta *}$$ are $$\eta $$-conjugate, $$\eta $$-Hermitian of X, respectively. We also present the solvability conditions and expression of the unique solution X or $$X=\pm X^{\star }$$.

On a new generalization of split Pell quaternions

On a new generalization of split Pell quaternions

Orbits in nonsupersymmetric magic theories

We determine and classify the electric-magnetic duality orbits of fluxes supporting asymptotically flat, extremal black branes in [Formula: see text] space–time dimensions in the so-called nonsupersymmetric magic Maxwell–Einstein theories, which are consistent truncations of maximal supergravity and which can be related to Jordan algebras (and related Freudenthal triple systems) over the split complex numbers [Formula: see text] and the split quaternions [Formula: see text]. By studying the stabilizing subalgebras of suitable representatives, realized as bound states of specific weight vectors of the corresponding representation of the electric-magnetic duality symmetry group, we obtain that, as for the case of maximal supergravity, in magic nonsupersymmetric Maxwell–Einstein theories there is no splitting of orbits, namely there is only one orbit for each nonmaximal rank element of the relevant Jordan algebra (in [Formula: see text] and 6) or of the relevant Freudenthal triple system (in [Formula: see text]).

Least squares X=±Xη* solutions to split quaternion matrix equation AXAη*=B

In the paper, the split quaternion matrix equation AXAη*=B is considered, where the operator Aη* is the η‐conjugate transpose of A, where η∈{i,j,k}. We propose some new real representations, which well exploited the special structures of the original matrices. By using this method, we obtain the necessary and sufficient conditions for AXAη*=B to have X=±Xη* solutions and derive the general expressions of solutions when it is consistent. In addition, we also derive the general expressions of the least squares X=±Xη* solutions to it in case that this matrix equation is not consistent.

Elementary Transformation and its Applications for Split Quaternion Matrices

In this paper, the elementary transformation is discussed for split quaternion matrices and the upper triangulation process is given. Then two new determinants are defined by means of real and complex representation matrices and the sufficient condition for the existence of LU decomposition is obtained. Finally, the inverse of the split quaternion matrix is studied and its existence condition and computation method are given.

Split Jacobsthal and Jacobsthal-Lucas Quaternions

In this paper, we introduce split Jacobsthal and split Jacobsthal-Lucas quaternions. We obtain generating functions and Binet’s formulas for these quaternions. We also investigate some properties of them.

Split Octonionic Cauchy Integral Formula

Cauchy integral formulas for the quaternions have been known since 1926. This result has previously been extended to the octonions and split quaternions. These techniques are extended to the split octonions. Thus Cauchy integral formulas are now known for all composition algebras over the real numbers.

Consistency of Split Quaternion Matrix Equations $$AX^{\star }-XB=CY+D$$ and $$X-AX^\star B=CY+D$$

In this paper, the split quaternion matrix equations $$AX^{\star }-XB=CY+D$$ and $$X-AX^{\star }B=CY+D$$ are considered, where $$X^\star $$ is X or the $$\phi $$ -conjugate of X, $$\phi \in \{i, j, k\}$$ . We design some new real representations of a split quaternion matrix, which enable us to convert $$\phi $$ -conjugate matrix equations into some real matrix equations. By using this method, we show that the original split quaternion matrix equation is consistent if and only if its corresponding real matrix equation is consistent. Moreover, we formulate the split quaternion solutions by the real solutions of its corresponding real matrix equation. As applications, we derive the necessary and sufficient conditions for the existence of solutions to the above two split quaternion matrix equations and their special cases $$AX^{\star }-XB=C$$ and $$X-AX^{\star }B=C$$ . We also obtain some conditions for the mentioned four split matrix equations to be uniquely solvable.

Iterative Methods for Least Squares Problem in Split Quaternionic Mechanics

Iterative Methods for Least Squares Problem in Split Quaternionic Mechanics

Real and Hyperbolic Matrices of Split Semi Quaternions

The aim of this paper is to investigate split semi quaternion matrices. To verify this, we first examine matrices with real split semi quaternion entries as a pair of hyperbolic matrices. Then, we present hyperbolic split semi quaternions and their matrices. Finally, for a hyperbolic split semi quaternion, we search some properties of its $$2\times 2$$ matrix representation and we express $$4\times 4$$ hyperbolic matrix and $$8\times 8$$ real matrix representations.

$${2\times 2}$$ 2 × 2 Matrix Representation Forms and Inner Relationships of Split Quaternions

This paper proposes a \({2\times 2}\) real matrix isomorphic representation form of split quaternions and a \({2m\times 2n}\) real matrix isomorphic representation form of \({m\times n}\) split quaternion matrices. In particular, we highlighted the inner relationships among the existing different \({2\times 2}\) matrix representation forms of split quaternions. Furthermore, we studied the inner relationships among different \({2\times 2}\) matrix representation forms of Hamilton quaternions as an application. The forms and relationships discussed in this paper can simplify the computation and make split quaternions get extensive applications.