in this paper, lie group structure and lie algebra structure of unit complex 3-sphere     are studied. in order to do this, adjoint representations of unit biquaternions (complexified quaternions) are obtained. also, a correspondence between the elements of     and the special complex unitary matrices    (2) is given by expressing biquaternions as 2-dimensional bicomplex numbers    .  the relat...

Since quaternions have isomorphic representations in matrix form we investigate various well known matrix decompositions for quaternions.

8 Vectors and Quaternions 145 8.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8.2 Displacement Vectors . . . . . . . . . . . . . . . . . . . . . . . 145 8.3 The Parallelogram Law of Vector Addition . . . . . . . . . . . 146 8.4 The Length of Vectors . . . . . . . . . . . . . . . . . . . . . . 147 8.5 Scalar Multiples of Vectors . . . . . . . . . . . . . . . . . . ....

8 Vectors and Quaternions 40 8.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 8.2 Displacement Vectors . . . . . . . . . . . . . . . . . . . . . . . 40 8.3 The Parallelogram Law of Vector Addition . . . . . . . . . . . 41 8.4 The Length of Vectors . . . . . . . . . . . . . . . . . . . . . . 42 8.5 Scalar Multiples of Vectors . . . . . . . . . . . . . . . . . . . . 43...

Abstract Quaternions are a four-dimensional hypercomplex number system discovered by Hamilton in 1843 and next intensively applied mathematics, modern physics, computer graphics other fields. After the discovery of quaternions, modified quaternions were also defined such way that commutative property multiplication is possible. That called as studied used for example signal processing. In this ...

Over the last decades quaternions have become a crucial and very successful tool for attitude representation in robotics and aerospace. However, there is a major problem that is continuously causing trouble in practice when it comes to exchanging formulas or implementations: there are two quaternion multiplications in common use, Hamilton’s multiplication and its flipped version, which is often...

5 Abstract Algebra 1 5.1 Binary Operations on Sets . . . . . . . . . . . . . . . . . . . . 1 5.2 Commutative Binary Operations . . . . . . . . . . . . . . . . 2 5.3 Associative Binary Operations . . . . . . . . . . . . . . . . . . 2 5.4 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5.5 The General Associative Law . . . . . . . . . . . . . . . . . . 4 5.6 Identity elements...

In this paper, real matrix representations of split quaternions are examined in terms of the casual character of quaternion. Then, we give De-Moivre’ s formula for real matrices of timelike and spacelike split quaternions, separately. Finally, we state the Euler theorem for real matrices of pure split quaternions.

9 Vectors and Quaternions 51 9.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 9.2 Displacement Vectors . . . . . . . . . . . . . . . . . . . . . . . 51 9.3 The Parallelogram Law of Vector Addition . . . . . . . . . . . 52 9.4 The Length of Vectors . . . . . . . . . . . . . . . . . . . . . . 53 9.5 Scalar Multiples of Vectors . . . . . . . . . . . . . . . . . . . . 54...

4 Vectors and Quaternions 47 4.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Displacement Vectors . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 The Parallelogram Law of Vector Addition . . . . . . . . . . . 49 4.4 The Length of Vectors . . . . . . . . . . . . . . . . . . . . . . 50 4.5 Scalar Multiples of Vectors . . . . . . . . . . . . . . . . . . . . 51...