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Math Help - Self adjoint and trace

  1. #1
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    [SOLVED]Self adjoint and trace

    Hi ALL !!!

    First, apology my english !!

    Look this problem:

    A is a self adjoint operator such that tr(A^2) = 0. Proof that A = 0.

    I thought use this property:

    <A,B> = tr(B*A)

    Because A is a self adjoint operator then I'll have that:

    <A,A> = tr(A*A) = tr(AA) = tr(A^2) = 0 => <A,A> = 0 => A = 0.

    But, I don't know how show that <A,B> = tr(B*A).

    Some one have some idea ???

    Thanks a lot !!
    Last edited by Borseti; December 1st 2011 at 09:16 AM.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Self adjoint and trace

    Quote Originally Posted by Borseti View Post
    I thought use this property: <A,B> = tr(B*A)
    Why? If A\in \mathbb{C}^{n\times n} is the matrix corresponding to a self adjoint operator then, A is similar to a diagonal matrix D=\textrm{diag}\;(\lambda_1,\ldots,\lambda_n)\in \mathbb{R}^{n\times n} . Now, use that similar matrices have the same trace.
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  3. #3
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    Re: Self adjoint and trace

    following up on previous post:

    A is similar to diagonal matrix A' whose diagonal elements are the eigenvalues of A. A'^2 = 0 so the sum of the squares of the eigenvalues of A are zero so the eigenvalues of A are zero so A is zero. (Trace is an invariant).

    EDIT: Argument invalid unless you show TrA^2=TrD^2
    Last edited by Hartlw; December 1st 2011 at 08:09 AM.
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  4. #4
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    Re: Self adjoint and trace

    Nice solution !!

    Thanks a lot !!!
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  5. #5
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    Re: Self adjoint and trace

    Missing step in previous argument:

    D=(P^-1)AP
    D^2 = [(P^-1)AP][(P^-1)AP] = (P^-1)(A^2)P

    Therefore D^2 similar to A^2 and TrD^2 = TrA^2 = 0.
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  6. #6
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    Re: Self adjoint and trace

    Thanks, I thought the same thing !!!
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  7. #7
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    Re: Self adjoint and trace

    To finish off previous argument, if A and B are similar, Ax=Bx all x. So Ax=Dx=0 all x => A=0.

    FrenandoRevilla took first step by pointing out a self-adjoint matrix is diagonalizable.
    Last edited by Hartlw; December 1st 2011 at 09:29 AM.
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  8. #8
    MHF Contributor Drexel28's Avatar
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    Re: Self adjoint and trace

    While I do like Fernando's solution, it's clearly the best, I do think your made a good observation. Namely, the map A\mapsto \tex{tr}(AA^\ast) defines a norm (the Frobenius norm or Hilbert-Schmidt norm--the name's to taste) and so A=0\Leftrightarrow \|A\|=0\Leftrightarrow 0=\text{tr}(AA^\ast)=\text{tr}(A^2).
    Last edited by Drexel28; December 3rd 2011 at 02:24 PM.
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  9. #9
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    Re: Self adjoint and trace

    Neither you nor Fernando solved the problem, or even came close.

    The solution is as I gave it, which I summarize here

    Given self-adjoint A and TrA^2=0. Prove A=0.

    1) A is self-adjoint therefore diagonalizable, D=PAP^-1 and similar to A.

    2) D^2 = P(A^2)P^-1 therefore D^2 is similar to A^2 and their trace is equal.

    3) TrD^2 = TrA^2 =0. Trace D^2 is sum of squares of eigenvalues so they are all 0 and D=0.

    4) If A and B are similar Ax=Bx all B, so Ax = Dx = 0 all x, so A=0

    Edited to change a typo in 4: "all x," not "all D." While I'm here, might as well improve step 4) which is correct but overly sophisticated. A = (P^-1)DP = 0.
    Last edited by Hartlw; December 3rd 2011 at 11:48 AM.
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  10. #10
    MHF Contributor Drexel28's Avatar
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    Lightbulb Re: Self adjoint and trace

    Quote Originally Posted by Hartlw View Post
    Neither you nor Fernando solved the problem, or even came close.
    I want to know I had to rewrite this, because my initial post was, put lightly, curt.

    I think you need to reexamine the situation.
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  11. #11
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    Irrelevant Premise Proof

    Irrelevant Premise Proof: The Premise is obviously true but irrelevant. Ex:

    1) The sky is blue
    2) 3 is an integer
    Therefore: sqrt2 is rational

    Advanced Irrelative Premise Proof: The premises are unnecessarily obtuse. If carefully crafted, they can contain unproven assumptions or need not even be true.

    These proofs are rock solid if an attempt is made to question the truth of the premise (the sky is blue).
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