PlanetPhysics/Example of a Matrix Commutator

Calculate the commutator $$[A,B]$$ of the matrices $$A = \left[ \begin{matrix} 5 & 3i & -i \\ 1 & 0 & 2 \\ i & 2i & -1 \end{matrix} \right] \,\,\, B = \left[ \begin{matrix} 1 & 0 & 1 \\ -2 & 2i & 3 \\ 3i & 1 & -1 \end{matrix} \right] $$

Ans.

The commutator is calculated by definition as

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

Carrying out the first matrix multiplication gives

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

and the second multiplication is

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

Finally, subtracting the two yields

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

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