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 geometric description of the multiplication of complex numbers can also be expressed in terms of [[rotation matrix|rotation matrices]] by using this correspondence between complex numbers and such matrices. The action of the matrix on a vector {{math|(''x'', ''y'')}} corresponds to the multiplication of {{math|''x'' + ''iy''}} by {{math|''a'' + ''ib''}}. In particular, if the determinant is {{math|1}}, there is a real number {{mvar|t}} such that the matrix has the form 

<math display=block>\begin{pmatrix}
  \cos t & - \sin t   \\
  \sin t & \;\; \cos t
\end{pmatrix}.</math>
In this case, the action of the matrix on vectors and the multiplication by the complex number <math>\cos t+i\sin t</math> are both the [[rotation (mathematics)|rotation]] of the angle {{mvar|t}}.