Editing Split-complex number (section)

==Matrix representations==
One can easily represent split-complex numbers by [[matrix (mathematics)|matrices]]. The split-complex number <math>z = x + jy</math> can be represented by the matrix <math>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 squared modulus of {{mvar|z}} is given by the [[determinant]] of the corresponding matrix.

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

<math display=block>m = \begin{pmatrix}a & c \\ b & -a \end{pmatrix}</math>

which square to the identity matrix satisfy <math>a^2 + bc = 1 .</math>
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>{{Better source needed|reason=Wikibooks not a reliable source ([[WP:USERGENERATED]]) |date=May 2023}}

The number <math>z = x + jy</math> can be represented by the matrix &nbsp;<math>x\ I + y\ m .</math>