Complex number: Difference between revisions

Revision as of 00:44, 17 October 2006

Revision as of 01:10, 22 October 2006

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a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to −1.

The absolute value of a complex number expressed as a matrix is equal to the [[square root]] of the [[determinant]] of that matrix.
:<math> |z|^2 =
\begin{vmatrix}
a & b \\
-b & a
\end{vmatrix}
= (a^2) - ((-b)(b)) = a^2 + b^2
</math>
If the matrix is viewed as a transformation of a plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number ''z'' corresponds to the transformation which rotates through the same angle as ''z'' but in the opposite direction, and scales in the same manner as ''z''; this can be described by the [[transpose]] of the matrix corresponding to ''z''.

If the matrix elements are themselves complex numbers, then the resulting algebra is that of the [[quaternions]]. In this way, the matrix representation can be seen as a way of expressing the [[Cayley-Dickson construction]] of algebras.