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Mathematics-Online course: Linear Algebra - Matrices - Matrix Operations

Commutator

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The commutator of two $ n\times n$-matrices is defined by

$\displaystyle [A,B] = AB - BA $

and, generally, it does not equal zero.
Composing two reflections $ s_1$ and $ s_2$ across different axes $ g_1$ and $ g_2$, resp., the order is relevant.

In the upper part of the figure the single reflections are depicted. In the lower part of the figure the double reflection $ s_2\circ s_1$ is shown on the left and the double reflection $ s_1\circ s_2$ is pictured on the right. The respective results are rotations by the double of the angle between the lines, however, in different directions.

\includegraphics[width=\linewidth]{b_doppelspiegelung_bild}

If we choose the first angle bisector as $ g_1$ and the $ y$-axis as $ g_2$, then the mappings have the following matrix representations:

$\displaystyle s_1:\quad v=S_1u=\left(\begin{array}{rr}0 & 1 \\ 1 & 0 \end{array}\right)u\,, $

$\displaystyle s_2:\quad v=S_2u\left(\begin{array}{rr}-1 & 0 \\ 0 & 1 \end{array}\right)u\,. $

The double reflections have the representations

$\displaystyle s_2\circ s_1:\quad w=S_2S_1u$ $\displaystyle =$ $\displaystyle \left(\begin{array}{rr}-1 & 0 \\ 0 & 1 \end{array}\right) \left(\begin{array}{rr}0 & 1 \\ 1 & 0 \end{array}\right)u$  
  $\displaystyle =$ $\displaystyle \left(\begin{array}{rr}0 & -1 \\ 1 & 0 \end{array}\right)u\,,$  


$\displaystyle s_1\circ s_2:\quad w=S_1S_2u$ $\displaystyle =$ $\displaystyle \left(\begin{array}{rr}0 & 1 \\ 1 & 0 \end{array}\right) \left(\begin{array}{rr}-1 & 0 \\ 0 & 1 \end{array}\right)u$  
  $\displaystyle =$ $\displaystyle \left(\begin{array}{rr}0 & 1 \\ -1 & 0 \end{array}\right)u\,.$  

In this case the commutator is

$\displaystyle [S_1,S_2] = \left(\begin{array}{rr}0 & 1 \\ -1 & 0 \end{array}\ri... ...{array}\right)= \left(\begin{array}{rr}0 & 2 \\ -2 & 0 \end{array}\right)\,. $

(Authors: Burkhardt/Höllig/Hörner)
  automatically generated 4/21/2005