Conjugation, The Norm, and Reciprocal
Conjugation of quaternions is analogous to conjugation of complex numbers and to transposition (also known as reversal) of elements of Clifford algebras. To define it, let q = a + bi + cj + dk be a quaternion. The conjugate of q is the quaternion a − bi − cj − dk. It is denoted by q∗, q, qt, or . Conjugation is an involution, meaning that it is its own inverse, so conjugating an element twice returns the original element. 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∗.
Unlike the situation in the complex plane, the conjugation of a quaternion can be expressed entirely with multiplication and addition:
Conjugation can be used to extract the scalar and vector parts of a quaternion. The scalar part of p is (p + p∗)/2, and the vector part of p is (p − p∗)/2.
The square root of the product of a quaternion with its conjugate is called its norm and is denoted ||q||. (Hamilton called this quantity the tensor of q, but this conflicts with modern usage. See tensor.) It has the formula
This is always a non-negative real number, and it is the same as the Euclidean norm on H considered as the vector space R4. Multiplying a quaternion by a real number scales its norm by the absolute value of the number. That is, if α is real, then
This is a special case of the fact that the norm is multiplicative, meaning that
for any two quaternions p and q. Multiplicativity is a consequence of the formula for the conjugate of a product. Alternatively multiplicativity follows directly from the corresponding property of determinants of square matrices and the formula
where i denotes the usual imaginary unit.
This norm makes it possible to define the distance d(p, q) between p and q as the norm of their difference:
This makes H into a metric space. Addition and multiplication are continuous in the metric topology.
A unit quaternion is a quaternion of norm one. Dividing a non-zero quaternion q by its norm produces a unit quaternion Uq called the versor of q:
Every quaternion has a polar decomposition q = ||q|| Uq.
Using conjugation and the norm makes it possible to define the reciprocal of a quaternion. The product of a quaternion with its reciprocal should equal 1, and the considerations above imply that the product of q and q∗/||q||2 (in either order) is 1. So the reciprocal of q is defined to be
This makes it possible to divide two quaternions p and q in two different ways. That is, their quotient can be either p q −1 or q −1 p. The notation p/q is ambiguous because it does not specify whether q divides on the left or the right.
Read more about this topic: Quaternions
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