Editing Quaternion (section)

== Conjugation, the norm, and reciprocal ==
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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 <math>q = a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k </math> be a quaternion. The '''[[Conjugate (algebra)|conjugate]]''' of {{mvar|q}} is the quaternion <math> q^* = a - b\,\mathbf i - c\,\mathbf j - d\,\mathbf k </math>. It is denoted by {{math|''q''<sup>∗</sup>}}, ''q<sup>t</sup>'', <math>\tilde q</math>, or {{overline|''q''}}.<ref name="SeeHazewinkel"/> Conjugation is an [[involution (mathematics)|involution]], meaning that it is its own [[inverse function|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 {{mvar|p}} and {{mvar|q}} are quaternions, then {{math|1=(''pq'')<sup>∗</sup> = ''q''<sup>∗</sup>''p''<sup>∗</sup>}}, not {{math|''p''<sup>∗</sup>''q''<sup>∗</sup>}}.

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The conjugation of a quaternion, in contrast to the complex setting, can be expressed with multiplication and addition of quaternions:

<math display=block>
q^* = - \tfrac{1}{2} (q + \mathbf i \,q \,\mathbf i + \mathbf j \,q \,\mathbf j + \mathbf k \,q \,\mathbf k).
</math>

Conjugation can be used to extract the scalar and vector parts of a quaternion. The scalar part of {{mvar|p}} is {{math|{{sfrac|1|2}}(''p'' + ''p''<sup>∗</sup>)}}, and the vector part of {{mvar|p}} is {{math|{{sfrac|1|2}}(''p'' − ''p''<sup>∗</sup>)}}.

{{Anchor|Norm}}The [[square root]] of the product of a quaternion with its conjugate is called its [[norm (mathematics)|''norm'']] and is denoted {{math|{{norm|''q''}}}} (Hamilton called this quantity the [[Tensor of a quaternion|''tensor'' of ''q'']], but this conflicts with the modern meaning of "[[tensor]]"). In formulas, this is expressed as follows:

<math display=block>\lVert q \rVert = \sqrt{qq^*} = \sqrt{q^*q} = \sqrt{a^2 + b^2 + c^2 + d^2}</math>

This is always a non-negative real number, and it is the same as the Euclidean norm on <math>\mathbb H</math> considered as the vector space <math>\mathbb R^4</math>. Multiplying a quaternion by a real number scales its norm by the absolute value of the number. That is, if {{mvar|α}} is real, then

<math display=block>\lVert\alpha q\rVert = \left| \alpha\right|\,\lVert q\rVert.</math>

This is a special case of the fact that the norm is ''multiplicative'', meaning that

<math display=block>\lVert pq \rVert = \lVert p \rVert\,\lVert q \rVert</math>

for any two quaternions {{mvar|p}} and {{mvar|q}}. Multiplicativity is a consequence of the formula for the conjugate of a product.
Alternatively it follows from the identity

<math display=block> \det \begin{pmatrix}
a + i b & i d + c \\
 i d - c & a - i b
\end{pmatrix} = a^2 + b^2 + c^2 + d^2,</math>

(where {{mvar|i}} denotes the usual [[imaginary unit]]) and hence from the multiplicative property of [[determinant]]s of square matrices.

This norm makes it possible to define the '''distance''' {{math|''d''(''p'', ''q'')}} between {{mvar|p}} and {{mvar|q}} as the norm of their difference:

<math display=block>d(p, q) = \lVert p - q \rVert.</math>

This makes <math>\mathbb H</math> a [[metric space]].
Addition and multiplication are [[continuous function (topology)|continuous]] in regard to the associated [[Metric space#Open and closed sets, topology and convergence|metric topology]].
This follows with exactly the same proof as for the real numbers <math>\mathbb R</math> from the fact that <math>\mathbb H</math> is a normed algebra.

=== Unit quaternion ===
{{Main|Versor}}
A '''unit quaternion''' is a quaternion of norm one. Dividing a nonzero quaternion {{mvar|q}} by its norm produces a unit quaternion {{math|'''U'''''q''}} called the ''[[versor]]'' of {{mvar|q}}:

<math display=block>\mathbf{U}q = \frac{q}{\lVert q\rVert}.</math>

Every nonzero quaternion has a unique [[polar decomposition]] <math> q = \lVert q \rVert \cdot \mathbf{U} q, </math> while the zero quaternion can be formed from any unit quaternion.

Using conjugation and the norm makes it possible to define the [[Multiplicative inverse|reciprocal]] of a nonzero quaternion. The product of a quaternion with its reciprocal should equal 1, and the considerations above imply that the product of <math>q</math> and <math>q^* / \left \Vert q \right \| ^2</math> is 1 (for either order of multiplication). So the ''[[reciprocal (mathematics)|reciprocal]]'' of {{mvar|q}} is defined to be

<math display=block>q^{-1} = \frac{q^*}{\lVert q\rVert^2}.</math>

Since the multiplication is non-commutative, the quotient quantities {{math|''p q''<sup>−1</sup>}} or {{math|''q''<sup>−1</sup>''p''}}&nbsp; are different (except if {{mvar|p}} and {{mvar|q}} are scalar multiples of each other or if one is a scalar): the notation {{math|{{sfrac|''p''|''q''}}}} is ambiguous and should not be used.