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Arithmetic and Geometric Mean

  1. Apr 30, 2012 #1
    Hi,

    I have the following equation:

    [tex]\gamma=\frac{1}{\frac{1}{N}\sum_{n=1}^N|\lambda_n|^{-2}}[/tex]

    where lambdas are the eigenvalues of an N-by-N circulant matrix A.

    I used two properties to bound the above equation:

    [tex]\frac{1}{N}\sum_{n=1}^N|\lambda_n|^{-2}\geq\left(\prod_{n=1}^N|\lambda_n|^{-2}\right)^{1/N}[/tex]

    [tex]\sum_{n=1}^N|\lambda_n|^{-2}\leq\left(\sum_{n=1}^N|\lambda_n|^{2}\right)^{-1}[/tex]

    Are these two bounds correct?

    Thanks
     
  2. jcsd
  3. Apr 30, 2012 #2

    mathman

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    The first one is a a statement that the arithmetic mean bounds the geometric mean (true).

    The second is false - simple example: 2 terms on the left 1 + 2 = 3. Right side is 1/(1+1/2) = 2/3.
     
  4. May 1, 2012 #3
    But we have the identity for the trace that:

    [tex]\sum_{n=1}^N\lambda_n^k=\text{Tr}\left(\mathbf{A}^k\right)\leq\left[\text{Tr}\left(\mathbf{A}\right)\right]^k=\left(\sum_{n=1}^N\lambda_n\right)^k[/tex]

    or it just works for k>=1?
     
  5. May 1, 2012 #4

    mathman

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    I am not familiar with the trace equation, but as my example shows, what you propose is just wrong. It may be that your guess is correct, k is positive integer.
     
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