Editing Quaternion (section)

=== 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''' }}.