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==Matrix representations==
One can easily represent split-complex numbers by [[matrix (mathematics)|matrices]]. The split-complex number
<math >z display= "block" x + jy</math> can be represented by the matrix <math>z \mapsto \begin{pmatrix}x & y \\ y & x\end{pmatrix}.</math> ▼<math display="block">z = x + jy</math>
can be represented by the matrix
▲<math display="block">z \mapsto \begin{pmatrix}x & y \\ y & x\end{pmatrix}.</math>
Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The modulus of {{mvar|z}} is given by the [[determinant]] of the corresponding matrix. In this representation, split-complex conjugation corresponds to multiplying on both sides by the matrix
<math display="block">C = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}.</math>
In fact there are many representations of the split-complex plane in the four-dimensional [[ring (mathematics)|ring]] of 2x2 real matrices. The real multiples of the [[identity matrix]] form a [[real line]] in the matrix ring M(2,R). Any hyperbolic unit ''m'' provides a [[basis (linear algebra)|basis]] element with which to extend the real line to the split-complex plane. The matrices
For any real number {{mvar|a}}, a hyperbolic rotation by a [[hyperbolic angle]] {{mvar|a}} corresponds to multiplication by the matrix
:<math>m = \begin{pmatrix}a & c \\ b & -a \end{pmatrix}</math> which square to the identity matrix satisfy <math>a^2 + bc = 1 .</math>
<math display="block">\begin{pmatrix}
For example, when ''a'' = 0, then (''b,c'') is a point on the standard hyperbola. More generally, there is a hypersurface in M(2,R) of hyperbolic units, any one of which serves in a basis to represent the split-complex numbers as a [[subring]] of M(2,R).<ref>{{wikibooks-inline|Abstract Algebra/2x2 real matrices}}</ref>
\cosh a & \sinh a \\
\sinh a & \cosh a
\end{pmatrix}.</math>
The number <math>z = x + jy</math> can be represented by the matrix <math>x\ I + y\ m .</math>
[[File:Commutative diagram split-complex number 2.svg|right|200px|thumb|This [[commutative diagram]] relates the action of the hyperbolic versor on {{mvar|D}} to squeeze mapping {{mvar|σ}} applied to {{tmath|\R^2}}]]
The diagonal basis for the split-complex number plane can be invoked by using an ordered pair {{math|(''x'', ''y'')}} for <math>z = x + jy</math> and making the mapping
<math display="block">(u, v) = (x, y) \begin{pmatrix}1 & 1 \\1 & -1\end{pmatrix} = (x, y) S ~.</math>
Now the quadratic form is <math>uv = (x + y)(x - y) = x^2 - y^2 ~.</math> Furthermore,
<math display="block">(\cosh a, \sinh a) \begin{pmatrix}
1 & 1 \\
1 & -1
\end{pmatrix} = \left(e^a, e^{-a}\right)</math>
so the two [[one-parameter group|parametrized]] hyperbolas are brought into correspondence with {{mvar|S}}.
The [[Group action (mathematics)|action]] of [[versor#Hyperbolic versor|hyperbolic versor]] <math>e^{bj} \!</math> then corresponds under this linear transformation to a [[squeeze mapping]]
<math display="block">\sigma: (u, v) \mapsto \left(ru, \frac{v}{r}\right),\quad r = e^b ~.</math>
There are many different representations of split-complex numbers in the 2×2 real matrices. In fact, every matrix whose square is the identity matrix gives such a representation.<ref>{{Wikibooks-inline|Abstract Algebra/2x2 real matrices}}</ref>
The above diagonal representation represents the [[Jordan canonical form]] of the matrix representation of the split-complex numbers. For a split-complex number {{math|1=''z'' = (''x'', ''y'')}} given by the following matrix representation:
<math display="block">Z = \begin{pmatrix}x & y \\ y & x\end{pmatrix} </math>
its Jordan canonical form is given by:
<math display="block">J_z = \begin{pmatrix}x + y & 0 \\ 0 & x - y\end{pmatrix} ~,</math>
where <math>Z = SJ_z S^{-1}\, ,</math> and
<math display="block">S = \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} ~.</math>
==History==
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