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Linear Algebra - Hermitian matrices

  1. Oct 16, 2005 #1
    Sorry if this is in the wrong section. I just want to check my answer since I've been going through the exam.


    Given that A is an n × n matrix and I is the n × n identity matrix, select all the correct responses below.
    A Every diagonalisable matrix is normal.
    B If A is Hermitian, then [tex]A^TA[/tex] is also Hermitian.
    C If all eigenvalues of A have algebraic multiplicity 1, then it is diagonalisable.
    D If A is Hermitian, then A + I is always invertible.
    E If B = A*A, then B is Hermitian.
    F None of the above

    I know that A is wrong because it is only every unitarily diagonalisable matrix that is normal. I think B is correct because if it is Hermitian it means that A is equal to the complex conjugate of the transpose. So really A is a symmetric matrix and [tex]A^T=A=A^*[/tex]. Not too sure about C in the end. I think D is wrong because the Hermitian matrix doesn't have to be invertible to begin with. E is also wrong because you don't know enough information about the matrix B to make that statement. Obviously the selection then can't be "none of the above".

    I think I might have them right, but I would like to check. ALthough I don't actually know about C so if someone could explain that to me I'd be grateful.

    Thanks
     
  2. jcsd
  3. Oct 16, 2005 #2

    Gokul43201

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    No, A need not be symmetric. Where did you get that from ? It doesn't say anywhere that A is real.

    PS : You are correct that choice A (all diag matrices are normal) is wrong.
     
    Last edited: Oct 16, 2005
  4. Oct 16, 2005 #3
    Oh I just thought that any matrix where [tex]A^T=A[/tex] is called a symmetric matrix.
     
  5. Oct 16, 2005 #4

    Gokul43201

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    That's right, but where is that given in the question ?
     
  6. Oct 16, 2005 #5
    Ahh wait yeah I'm an idiot...I just took something out of the statement that wasn't there. It doesn't equal just A transpose. It's only the complex conjugate of the transpose. OK so B is actually wrong as well.
     
  7. Oct 16, 2005 #6

    Hurkyl

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    For the ones you think are wrong, have you tried coming up with a counterexample?
     
  8. Oct 18, 2005 #7
    haha thanks for that. I just tried some examples for E and found that it was true. Also found a proof in the notes that C was true so that was good. So yeah I found that C and E are the actual answers (I'm pretty damn sure about this)...can't believe how off my answers were at the beginning.

    Thanks again
     
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