Question

two marices A and B are hermiian find a necessary and sufficient condition that AB is also hemitian

Answer 1

*

Answer 2

That the matrices commute:
AB = BA.
It is necessary and sufficient.
Guide: proof of sufficiency is easy - just take conjugate adjoint over complex field. Then use the assumption of Hermiticity of each one.
Necessity - look at sufficiency and do something not far from it

Answer 3

This is the outline. Details elsewhere.

Answer 4

thx U.R.