The division algebra of real quaternions, as the only noncommutative normed division real algebra up to isomorphism of normed algebras, is of great importance. In this note, first we present a brief introduction to quaternion matrices and quaternion linear algebra. This, among other things, will help us present the counterpart of a theorem of Herman Auerbach in the setting of quaternions. More ...

Two studies [Ivry, R. B., Franz, E. A., Kingstone, A., & Johnston, J. C. The psychological refractory period effect following callosotomy: Uncoupling of lateralized response codes. Journal of Experimental Psychology: Human Perception and Performance, 24, 463-480, 1998; Pashler, H., Luck, S., Hillyard, S. A., Mangun, G. R., O'Brien, S., & Gazzaniga, M. S. Sequential operation of disconnected hem...

The theory of quaternions was introduced in the mid nineteenth century, and it found many applications in classical mechanics, quantum mechanics, and the theory of relativity. Quaternions were also later used in aerospace applications and flight simulators, particularly when inertial attitude referencing and related control schemes where employed. However, it is only in the recent past that gra...

The underwater glider has difficulty accessing the complex and narrow hadal trench for observation, which is affected by its limited regulation capability of pitch angle (?45°~45°). In this study, a compact attitude regulating mechanism proposed to extend range from ?90°to 90° install it on hadal-class Petrel-XPLUS. Subsequently, dynamics model Petrel-XPLUS established using dual quaternions so...

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 ...