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The conjugate of a product of two quaternions is the product of the conjugates in the reverse order. That is, if p and q are quaternions, then (pq) ∗ = q ∗ p ∗, not p ∗ q ∗. The conjugation of a quaternion, in stark contrast to the complex setting, can be expressed with multiplication and addition of quaternions:
Since unit quaternions are subject to two algebraic constraints, unit quaternions are standard to represent rigid transformations. [2] Similar to the way that rotations in 3D space can be represented by quaternions of unit length, rigid motions in 3D space can be represented by dual quaternions of unit length.
Multiplication of two real numbers, two imaginary numbers or a real number by an imaginary number in the classical notation system was the same operation. Multiplication of a scalar and a vector was accomplished with the same single multiplication operator; multiplication of two vectors of quaternions used this same operation as did ...
In the same way the quaternions can be defined by introducing abstract symbols i, j, k which satisfy the rules i 2 = j 2 = k 2 = i j k = −1 and the usual algebraic rules except the commutative law of multiplication (a familiar example of such a noncommutative multiplication is matrix multiplication).
The quaternions form a noncommutative ring, and therefore addition and multiplication can be defined for quaternionic matrices as for matrices over any ring. Addition. The sum of two quaternionic matrices A and B is defined in the usual way by element-wise addition: (+) = +. Multiplication.
Using the distributive property, these relations can be used to multiply any two hyperbolic quaternions. Unlike the ordinary quaternions, the hyperbolic quaternions are not associative . For example, ( i j ) j = k j = − i {\displaystyle (ij)j=kj=-i} , while i ( j j ) = i {\displaystyle i(jj)=i} .
The red cycle also reflects that i 2 = e, i 3 = i and i 4 = e. In group theory, the quaternion group Q 8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset {,,,,,} of the quaternions under multiplication.
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.