Editing Complex number (section)

===Matrix representation of complex numbers===<!-- .This section is linked from [[Cauchy-Riemann equations]] -->
Complex numbers {{math|''a'' + ''bi''}} can also be represented by {{math|2 × 2}} [[matrix (mathematics)|matrices]] that have the form
<!-- 
This definition with the minus sign in the upper right corner matches the article [[Rotation matrix]]. Please do not change it.
-->
<math display=block>
\begin{pmatrix}
  a &   -b  \\
  b & \;\; a
\end{pmatrix}.
</math>
Here the entries {{mvar|a}} and {{mvar|b}} are real numbers. As the sum and product of two such matrices is again of this form, these matrices form a [[subring]] of the ring of {{math|2 × 2}} matrices.

A simple computation shows that the map
<math display=block>a+ib\mapsto \begin{pmatrix}
  a &   -b  \\
  b & \;\; a
\end{pmatrix}</math>
is a [[ring isomorphism]] from the field of complex numbers to the ring of these matrices, proving that these matrices form a field. This isomorphism associates the square of the absolute value of a complex number with the [[determinant]] of the corresponding matrix, and the conjugate of a complex number with the [[transpose]] of the matrix.

The [[polar form]] representation of complex numbers explicitly gives these matrices as scaled [[rotation matrix|rotation matrices]].
<math display=block>r (\cos \theta + i \sin \theta)\mapsto \begin{pmatrix}
  r \cos \theta &   -r \sin \theta  \\
  r \sin \theta & \;\; r \cos \theta
\end{pmatrix}</math>
In particular, the case of {{math|1=''r'' = 1}}, which is <math>|a + ib| = \sqrt{a^2+b^2} = 1</math>, gives (unscaled) rotation matrices.