If A is a Hermitian matrix, and B is a complex matrix of same order as A, then $\displaystyle BAB^* $ is also a Hermitian matrix.

Anyone know how to prove this other than just writing it all out by hand?

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- Apr 21st 2013, 01:45 PMjpquinn91Hermitian matrix proof
If A is a Hermitian matrix, and B is a complex matrix of same order as A, then $\displaystyle BAB^* $ is also a Hermitian matrix.

Anyone know how to prove this other than just writing it all out by hand? - Apr 21st 2013, 06:34 PMmathguy25Re: Hermitian matrix proof
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.