Editing Polar decomposition (section)

==Alternative planar decompositions==
In the [[Cartesian plane]], alternative planar [[ring (mathematics)|ring]] decompositions arise as follows:
* If {{math|''x'' ≠ 0}}, {{math|1=''z'' = ''x''(1 + ε(''y''/''x''))}} is a polar decomposition of a [[dual number]] {{math|1=''z'' = ''x'' + ''yε''}}, where {{math|1=''ε''<sup>2</sup> = 0}}; i.e., ''ε'' is [[nilpotent]]. In this polar decomposition, the unit circle has been replaced by the line {{math|1=''x'' = 1}}, the polar angle by the [[slope]] ''y''/''x'', and the radius ''x'' is negative in the left half-plane.
* If {{math|''x''<sup>2</sup> ≠ ''y''<sup>2</sup>}}, then the [[unit hyperbola]] {{math|1=''x''<sup>2</sup> − ''y''<sup>2</sup> = 1}}, and [[conjugate hyperbola|its conjugate]] {{math|1=''x''<sup>2</sup> − ''y''<sup>2</sup> = −1}} can be used to form a polar decomposition based on the branch of the unit hyperbola through {{math|(1, 0)}}. This branch is parametrized by the [[hyperbolic angle]] ''a'' and is written <math display="block">
 \cosh a + j \sinh a = \exp(aj) = e^{aj},
</math> where {{math|1=''j''<sup>2</sup> = +1}}, and the arithmetic<ref>Sobczyk, G. (1995) "Hyperbolic Number Plane", ''[[College Mathematics Journal]]'' 26:268–280.</ref> of [[split-complex number]]s is used. The branch through {{math|(−1, 0)}} is traced by −''e''<sup>''aj''</sup>. Since the operation of multiplying by ''j'' reflects a point across the line {{math|1=''y'' = ''x''}}, the conjugate hyperbola has branches traced by ''je''<sup>''aj''</sup> or −''je''<sup>''aj''</sup>. Therefore a point in one of the quadrants has a polar decomposition in one of the forms: <math display="block">
 r e^{aj}, -re^{aj}, rje^{aj}, -rje^{aj}, \quad r > 0.
</math> The set {{math|{1, −1, ''j'', −''j''}<nowiki/>}} has products that make it isomorphic to the [[Klein four-group]]. Evidently polar decomposition in this case involves an element from that group.

Polar decomposition of an element of the [[algebra over a field|algebra]] M(2,&nbsp;R) of 2&nbsp;×&nbsp;2 real matrices uses these alternative planar decompositions since any planar [[subalgebra]] is isomorphic to dual numbers, split-complex numbers, or ordinary complex numbers.