Definition: A is Hermitian iff A* = A where A* = (A bar)^T

Remember the following facts

1. (AB)* = B* A*

2. (A*)* = A

Need to show that (BAB*)* = BAB*

By 1. we see that (BAB*)* = (B*)*(A*)(B*)

By 2. we see that (B*)*(A*)(B*) = B(A*)(B*) since (B*)* = B

Since A is Hermitian, A* = A. Thus, B(A*)(B*) = BAB*.

Therefore,

(BAB*)* = (B*)*(A*)(B*) = B(A*)(B*) = BAB*. Thus, BAB* is Hermitian.