Editing Split-quaternion (section)

== Representation as complex matrices ==
There is a representation of the split-quaternions as a [[unital associative algebra|unital associative subalgebra]] of the {{math|2×2}} matrices with [[complex number|complex]] entries. This representation can be defined by the [[algebra homomorphism]] that maps a split-quaternion {{math|''w'' + ''x''i + ''y''j + ''z''k}} to the matrix
: <math>\begin{pmatrix}w+xi& y+zi\\y-zi&w-xi\end{pmatrix}.</math>
Here, {{mvar|i}} ([[italic type|italic]]) is the [[imaginary unit]], not to be confused with the split quaternion basis element {{math|i}} ([[roman type|upright roman]]).

The image of this homomorphism is the [[matrix ring]] formed by the matrices of the form
: <math>\begin{pmatrix}u & v \\ v^* & u^* \end{pmatrix},</math>
where the superscript <math>^*</math> denotes a [[complex conjugate]].

This homomorphism maps respectively the split-quaternions {{math|i, j, k}} on the matrices
: <math>\begin{pmatrix}i & 0 \\0 &-i \end{pmatrix}, \quad\begin{pmatrix}0 & 1 \\1 &0 \end{pmatrix},\quad \begin{pmatrix}0 & i \\-i &0 \end{pmatrix}.</math>

The isomorphism of algebras is completed by use of [[matrix multiplication]] to verify the identities involving i, j, and k. For instance,
:<math>j k = \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix}0 & i \\ -i & 0 \end{pmatrix} = \begin{pmatrix}-i & 0 \\ 0 & i \end{pmatrix} = - i .</math>

It follows that for a split quaternion represented as a complex matrix, the conjugate is the matrix of the cofactors, and the norm is the determinant.

With the representation of split quaternions as complex matrices, the matrices of determinant {{math|1}} form the special unitary group [[SU(1,1)]], which is used to describe  [[hyperbolic motion#Disk model motions|hyperbolic motions]] of the [[Poincaré disk model]]  in [[hyperbolic geometry]].<ref>Karzel, Helmut & Günter Kist (1985) "Kinematic Algebras and their Geometries", in ''Rings and Geometry'', R. Kaya, P. Plaumann, and K. Strambach editors, pp. 437&ndash;509, esp 449,50, [[D. Reidel]] {{isbn|90-277-2112-2}}</ref>