Editing Dual number (section)

==Matrix representation==

The dual number <math>a + b \varepsilon</math> can be represented by the [[square matrix]] <math>\begin{pmatrix}a & b \\ 0 & a \end{pmatrix}</math>. In this representation the matrix <math>\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}</math> squares to the zero matrix, corresponding to the dual number <math>\varepsilon</math>.

Generally, if <math>\varepsilon</math> is a [[nilpotent element|nilpotent]] matrix, then ''B'' = {''x'' I + ''y'' <math>\varepsilon</math>: ''x, y'' real} is a [[subalgebra]] isomorphic to the algebra of dual numbers. In the case of 2x2 real matrices M(2,'''R'''), <math>\varepsilon</math> can be taken as any matrix of the form <math>\begin{pmatrix}a & b \\ c & -a \end{pmatrix}</math> with ''p'' = ''a''<sup>2</sup> + ''bc'' = 0.

The dual numbers are one of three isomorphism classes of real 2-algebras in M(2,'''R'''). When ''p'' > 0 the subalgebra ''B'' is isomorphic to [[split-complex number]]s, and when ''p'' < 0, ''B'' is isomorphic to the [[complex plane]].