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Math Help - Hermitian matrix proof

  1. #1
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    Hermitian matrix proof

    If A is a Hermitian matrix, and B is a complex matrix of same order as A, then  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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  2. #2
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    Re: 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.
    Thanks from jpquinn91
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