Question

Is the commutator of two hermition matrix again a hermition?

Answer 1

Answer:

Let A, B, be 2 hermitian matrices.

[A,B]*=(AB-BA)*

<=> [A,B]*=B*A*-A*B*

<=> [A,B]*=BA-AB

<=> [A,B]*=-[A,B]

So the commutator is skew-hermitian, not hermitian.