Editing Quaternion (section)

== Matrix representations ==
Just as complex numbers can be [[Matrix representation of complex numbers|represented as matrices]], so can quaternions. There are at least two ways of representing quaternions as matrices in such a way that quaternion addition and multiplication correspond to matrix addition and [[matrix multiplication]]. One is to use 2 × 2 complex matrices, and the other is to use 4 × 4 [[real number|real]] matrices. In each case, the representation given is one of a family of linearly related representations. These are [[injective function|injective]] [[ring homomorphism|homomorphism]]s from <math>\mathbb H</math> to the [[matrix ring]]s {{math|M(2,'''C''')}} and {{math|M(4,'''R''')}}, respectively.

=== 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>

=== Representation as real 4 × 4 matrices ===
Using 4 × 4 real matrices, that same quaternion can be written as

<math display=block>\begin{align}
\left[ \begin{array}{rrrr}
 a & -b & -c & -d \\
 b & a  & -d & c  \\
 c & d  & a  & -b \\
 d & -c & b  & a
\end{array} \right]
&= a
\left[ \begin{array}{rrrr}
 1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & 1
\end{array} \right]
+ b
\left[ \begin{array}{rrrr}
 0 & -1 & 0 & 0  \\
 1 & 0  & 0 & 0  \\
 0 & 0  & 0 & -1 \\
 0 & 0  & 1 & 0
\end{array} \right] \\[10mu]
&\qquad + c
\left[ \begin{array}{rrrr}
 0 & 0 & -1 & 0 \\
 0 & 0 & 0  & 1 \\
 1 & 0 & 0  & 0 \\
 0 & -1 & 0 & 0
\end{array} \right]
+ d
\left[ \begin{array}{rrrr}
 0 & 0 & 0  & -1 \\
 0 & 0 & -1 & 0  \\
 0 & 1 & 0  & 0  \\
 1 & 0 & 0  & 0
\end{array} \right].
\end{align}</math>

However, the representation of quaternions in {{math|M(4,'''R''')}} is not unique. For example, the same quaternion can also be represented as

<math display=block>\begin{align}
\left[ \begin{array}{rrrr}
 a  & d  & -b & -c \\
 -d & a  & c  & -b \\
 b  & -c & a  & -d \\
 c  & b  & d  & a
\end{array} \right]
&= a
\left[ \begin{array}{rrrr}
 1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & 1
\end{array} \right]
+ b
\left[ \begin{array}{rrrr}
 0 & 0 & -1 & 0  \\
 0 & 0 & 0  & -1 \\
 1 & 0 & 0  & 0  \\
 0 & 1 & 0  & 0
\end{array} \right] \\[10mu]
&\qquad + c
\left[ \begin{array}{rrrr}
 0 & 0  & 0 & -1 \\
 0 & 0  & 1 & 0  \\
 0 & -1 & 0 & 0  \\
 1 & 0  & 0 & 0
\end{array} \right]
+ d
\left[ \begin{array}{rrrr}
 0  & 1 & 0 & 0  \\
 -1 & 0 & 0 & 0  \\
 0  & 0 & 0 & -1 \\
 0  & 0 & 1 & 0
\end{array} \right].
\end{align}</math>

There exist 48&nbsp;distinct matrix representations of this form in which one of the matrices represents the scalar part and the other three are all skew-symmetric. More precisely, there are 48&nbsp;sets of quadruples of matrices with these symmetry constraints such that a function sending {{math|1, '''i''', '''j'''}}, and {{math|'''k'''}} to the matrices in the quadruple is a homomorphism, that is, it sends sums and products of quaternions to sums and products of matrices.<ref name="MatRep">{{cite journal |last1=Farebrother |first1=Richard William |last2=Groß |first2=Jürgen |last3=Troschke |first3=Sven-Oliver |title=Matrix representation of quaternions |journal=Linear Algebra and Its Applications |date=2003 |volume=362 |pages=251–255 |doi=10.1016/s0024-3795(02)00535-9 | doi-access=free }}</ref> 
In this representation, the conjugate of a quaternion corresponds to the [[transpose]] of the matrix. The fourth power of the norm of a quaternion is the [[determinant]] of the corresponding matrix. The scalar part of a quaternion is one quarter of the matrix trace. As with the 2 × 2 complex representation above, complex numbers can again be produced by constraining the coefficients suitably; for example, as block diagonal matrices with two 2 × 2 blocks by setting {{math|1=''c'' = ''d'' = 0}}.

Each 4×4 matrix representation of quaternions corresponds to a multiplication table of unit quaternions. For example, the last matrix representation given above corresponds to the multiplication table
{|class="wikitable" style="text-align:center"
|-
!width=15|×
!width=15|''a''
!width=15|''d''
!width=15|−''b''
!width=15|−''c''
|-
!''a''
|''a''
|''d''
|''−b''
|''−c''
|-
!''−d''
|''−d''
|''a''
|''c''
|''−b''
|-
!''b''
|''b''
| −''c''
|''a''
|−''d''
|-
!''c''
|''c''
|''b''
|''d''
|''a''
|-
|}

which is isomorphic — through <math>\{a \mapsto 1,\, b \mapsto i,\, c \mapsto j,\, d \mapsto k\}</math> — to

{|class="wikitable" style="text-align:center"
|-
!width=15|×
!width=15| 1 
!width=15|'''k'''
!width=15|−'''i'''
!width=15|−'''j'''
|-
!1 
|1 
|'''k'''
|−'''i'''
|−'''j'''
|-
!−'''k'''
|−'''k'''
|1
|'''j'''
|−'''i'''
|-
!'''i'''
|'''i'''
|−'''j'''
|1 
|−'''k'''
|-
!'''j'''
|'''j'''
|'''i'''
|'''k'''
|1
|-
|}

Constraining any such multiplication table to have the identity in the first row and column and for the signs of the row headers to be opposite to those of the column headers, then there are 3&nbsp;possible choices for the second column (ignoring sign), 2&nbsp;possible choices for the third column (ignoring sign), and 1 possible choice for the fourth column (ignoring sign); that makes 6&nbsp;possibilities. Then, the second column can be chosen to be either positive or negative, the third column can be chosen to be positive or negative, and the fourth column can be chosen to be positive or negative, giving 8 possibilities for the sign. Multiplying the possibilities for the letter positions and for their signs yields 48. Then replacing {{math|1}} with {{mvar|a}}, {{math|'''i'''}} with {{mvar|b}}, {{math|'''j'''}} with {{mvar|c}}, and {{math|'''k'''}} with {{mvar|d}} and removing the row and column headers yields a matrix representation of {{math|''a'' + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k''' }}.