Editing Quaternion (section)

=== Representation as complex 2 × 2 matrices ===
The quaternion {{math|''a'' + ''b'''''i''' + ''c'''''j''' + ''d'''''k'''}} can be represented using a complex 2 × 2 matrix as

<math display="block">\begin{bmatrix}
\phantom-a + bi & c + di \\
        -c + di & a - bi
\end{bmatrix}.</math>

This representation has the following properties:
* Constraining any two of {{mvar|b}}, {{mvar|c}} and {{mvar|d}} to zero produces a representation of complex numbers. For example, setting {{math|1=''c'' = ''d'' = 0}} produces a diagonal complex matrix representation of complex numbers, and setting {{math|1=''b'' = ''d'' = 0}} produces a real matrix representation.
* The norm of a quaternion (the square root of the product with its conjugate, as with complex numbers) is the square root of the [[determinant]] of the corresponding matrix.<ref>{{cite web |url=http://www.wolframalpha.com/input/?i=det+%7B{a%2Bb*i%2C+c%2Bd*i}%2C+{-c%2Bd*i%2C+a-b*i}%7D |title=[no title cited; determinant evaluation] |website=Wolframalpha.com}}</ref>
* The scalar part of a quaternion is one half of the [[matrix trace]].
* The conjugate of a quaternion corresponds to the [[conjugate transpose]] of the matrix.
* By restriction this representation yields a [[group isomorphism]] between the subgroup of unit quaternions and their image [[SU(2)]]. Topologically, the [[unit quaternion]]s are the 3-sphere, so the underlying space of SU(2) is also a 3-sphere. The group {{math|SU(2)}} is important for describing [[Spin (physics)|spin]] in quantum mechanics; see [[Pauli matrices]].
* There is a strong relation between quaternion units and Pauli matrices. The 2 × 2 complex matrix above can be written as <math>a I + b i \sigma_3 + c i \sigma_2 + d i \sigma_1</math>, so in this representation the quaternion units {{math|{{mset|1, '''i''', '''j''', '''k'''}}}} correspond to {{math|{{mset|'''I''', <math>i \sigma_3</math>,<math>i \sigma_2</math>, <math>i \sigma_1</math>}}}} = {{math|{{mset|'''I''', <math>\sigma_1 \sigma_2</math>,<math>\sigma_3 \sigma_1</math>, <math>\sigma_2 \sigma_3</math>}}}}. Multiplying any two Pauli matrices always yields a quaternion unit matrix, all of them except for −1. One obtains −1 via {{nowrap|{{math|1='''i'''<sup>2</sup> = '''j'''<sup>2</sup> = '''k'''<sup>2</sup> = '''i j k''' = −1}}}}; e.g. the last equality is <math display=block>\mathbf{i\;j\;k} = \sigma_1 \sigma_2 \sigma_3 \sigma_1 \sigma_2 \sigma_3 = -1.</math>
The representation in {{math|M(2,'''C''')}} is not unique.  A different convention, that preserves the direction of cyclic ordering between the quaternions and the Pauli matrices, is to choose <math display="block">
1 \mapsto \mathbf{I}, \quad \mathbf{i} \mapsto - i \sigma_1 = - \sigma_2 \sigma_3, \quad \mathbf{j} \mapsto - i \sigma_2  = - \sigma_3 \sigma_1, \quad \mathbf{k} \mapsto - i \sigma_3 = - \sigma_1 \sigma_2,
</math>{{pb}}This gives an alternative representation,<ref>eg Altmann (1986), ''Rotations, Quaternions, and Double Groups'', p. 212, eqn 5</ref> <math display="block">
a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k
\mapsto \begin{bmatrix} a - di & -c - bi \\ c - bi & \phantom-a + di \end{bmatrix}.
</math>