Quaternions - Conjugation, The Norm, and Reciprocal

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.