Past exam of the mathematics course of the University of Cambridge / 2014 / ia / Paper 1 / 8C / ii

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- Solution ii

Solution

The conjugate transpose reverses multiplication. If

A^{†} = - A

and

B^{†} = - B

, then

[A, B]^{†} = B^{†} A^{†} - A^{†} B^{†} = B A - A B = - [A, B] .

Thus the commutator is again a skew-Hermitian matrix. The order reversal is what produces the required minus sign.

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