This document provides a mathematical summary of quaternion algebra and calculus and how they relate to rotations and interpolation of rotations. The ideas are based on the article [1].

In this paper, we present quaternion matrix algebra techniques that can be used to process the eigen analysis of a color image. Applications of Principal Component Analysis (PCA) in image processing are numerous, and the proposed tools aim to give material for color image processing, that take into account their particular nature. For this purpose, we use the quaternion model for color images a...

The recently developed concepts of quaternion Fourier transform (QFT), quaternion convolution (QCV), and quaternion correlation, which are based on quaternion algebra, have been found to be useful for color image processing. However, the necessary computational algorithms and their complexity still need some attention. In this paper, we will develop efficient algorithms for QFT, QCV, and quater...

In the literature we find a variety of mathematical approaches for solving problems in robotics which we will review now briefly. Denavit and Hartenberg [60] introduced the mostly used kinematic notation for lower pair mechanisms based on matrix algebra, Walker [243] used the epsilon algebra for the treatment of the manipulator kinematics, Gu and Luh [99] utilized dual– matrices for computing t...

The classical Lie algebras over the complex numbers are all derived from associative algebras by defining a Lie bracket [A,B] = AB−BA, but the five exceptional Lie algebras are often defined directly as Lie algebras. In four cases, however, an alternative approach is available to construct both the algebra and the corresponding Lie group. It is well-known that the Lie group G2 is naturally defi...

As is well-known, the real quaternion division algebra H is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra O can not be algebraically isomorphic to any matrix algebras over the real number field R, because O is a non-associative algebra over R. However since O is an extension of H by the Cayley-Dickson process and is also finite-dimensional, som...

We won’t cover: • linear algebra, i.e., the underlying mathematics of matrices • numerical linear algebra, i.e., the algorithms used to manipulate matrices and solve linear equations • software for forming and manipulating matrices, e.g., Matlab, Mathematica, or Octave • how to represent and manipulate matrices, or solve linear equations, in computer languages such as C/C++ or Java • applicatio...