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Together with the imaginary unit i and the antilinear map k := ij, j equips V with the structure of a quaternionic vector space (i.e., V becomes a module over the division algebra of quaternions). From this point of view, quaternionic representation of a group G is a group homomorphism φ : G → GL( V , H ), the group of invertible quaternion ...
The real quaternion 1 is the identity element. The real quaternions commute with all other quaternions, that is aq = qa for every quaternion q and every real quaternion a. In algebraic terminology this is to say that the field of real quaternions are the center of this quaternion algebra.
The quaternion group has the unusual property of being Hamiltonian: Q 8 is non-abelian, but every subgroup is normal. [4] Every Hamiltonian group contains a copy of Q 8. [5] The quaternion group Q 8 and the dihedral group D 4 are the two smallest examples of a nilpotent non-abelian group.
Therefore, a direction-dependent derivative is necessary for functions of a quaternion variable. [4] [5] Considering the increment of polynomial function of quaternionic argument shows that the increment is a linear map of increment of the argument. [dubious – discuss] From this, a definition can be made:
The quaternion a + bi + cj + dk can be represented as the 2×2 complex matrix [ a + b i c + d i − c + d i a − b i ] . {\displaystyle {\begin{bmatrix}~~a+bi&c+di\\-c+di&a-bi\end{bmatrix}}.} This defines a map Ψ mn from the m by n quaternionic matrices to the 2 m by 2 n complex matrices by replacing each entry in the quaternionic matrix by ...
As a quaternion consists of two independent complex numbers, they form a four-dimensional vector space over the real numbers. The multiplication of quaternions is not quite like the multiplication of real numbers, though; it is not commutative – that is, if p and q are quaternions, it is not always true that pq = qp.
3D visualization of a sphere and a rotation about an Euler axis (^) by an angle of In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle about a fixed axis (called the Euler axis) that runs through the fixed point. [6]
In mathematics, a quaternion algebra over a field F is a central simple algebra A over F [1] [2] that has dimension 4 over F.Every quaternion algebra becomes a matrix algebra by extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension K of F, is isomorphic to the 2 × 2 matrix algebra over K.
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