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Logs, Traces, Matrices

  1. Aug 30, 2013 #1
    Trying to make sense of the following relation:

    [itex]\sum log d_{j} = tr log(D)[/itex]

    with D being a diagonalized matrix.

    Seems to imply the log of a diagonal matrix is the log of each element along the diagonal.

    Having a hard time convincing myself that is true, though
     
  2. jcsd
  3. Aug 30, 2013 #2
    One more:

    if [itex]M = A^{-1}DA[/itex],

    why is this true:

    [itex]tr A^{-1}log(D)A=tr\ log (M)[/itex]
     
  4. Aug 30, 2013 #3

    Office_Shredder

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    No, this is a fact about eigenvalues. If [itex] \lambda [/itex] is an eigenvalue of D, then [itex] \log(\lambda)[/itex] is an eigenvalue of [itex] \log(D)[/itex].

    This is usually first presented in the other direction, that if [itex] \lambda[/itex] is an eigenvalue of D, then [itex] e^{\lambda}[/itex] is an eigenvalue of [itex] e^D[/itex]. (and the eigenvector is the same). It's very easy to see this by writing out the power series definition of eD and applying it to the eigenvector of D
     
    Last edited: Aug 30, 2013
  5. Aug 30, 2013 #4
    ahh, thanks I should have known that.

    I figured out the second one too, so no help needed on that one now.
     
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