example of a matrix commutator

Calculate the commutator $ [A,B]$ of the matrices

$\displaystyle A =
\left[ \begin{array}{ccc}
5 & 3i & -i \\
1 & 0 & 2 \\
i & 2...
...egin{array}{ccc}
1 & 0 & 1 \\
-2 & 2i & 3 \\
3i & 1 & -1 \end{array} \right]
$

Ans.

The commutator is calculated by definition as

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

Carrying out the first matrix multiplication gives

$\displaystyle AB =
\left[ \begin{array}{ccc}
8-6i & -6-i & 5+10i \\
1+6i & 2 & -1 \\
-6i & -5 & 1+7i \end{array} \right]$

and the second multiplication is

$\displaystyle BA =
\left[ \begin{array}{ccc}
5+i & 5i & -1-i \\
-10+5i & 0 & -3+6i \\
1+14i & -9-2i & 6 \end{array} \right].$

Finally, subtracting the two yields

$\displaystyle [A,B] = AB-BA
\left[ \begin{array}{ccc}
3-7i & -6-6i & 6+11i \\
11+i & 2 & 2-6i \\
-1-20i & 4+2i & -5+7i \end{array} \right].$

Since the commutator is non-zero, we say that $ A$ and $ B$ do not commute.



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