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Arithmetic and Geometric Mean

  1. Apr 30, 2012 #1

    I have the following equation:


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

    I used two properties to bound the above equation:



    Are these two bounds correct?

  2. jcsd
  3. Apr 30, 2012 #2


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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:


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


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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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