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Taylor expansion for matrix logarithm

  1. Mar 29, 2013 #1
    A paper I'm reading states the that: for positive hermitian matrices A and B, the Taylor expansion of [itex]\log(A+tB)[/itex] at t=0 is

    [itex]\log(A+tB)=\log(A) + t\int_0^\infty \frac{1}{B+zI}A \frac{1}{B+zI} dz + \mathcal{O}(t^2).[/itex]

    However, there is no source or proof given, and I cannot seem to find a derivation of this identity anywhere! Any help would be appreciated. Thanks.
    Last edited: Mar 29, 2013
  2. jcsd
  3. Mar 29, 2013 #2

    I like Serena

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    Welcome to PF, Backpacker! :smile:

    I don't recognize your formula, but:

    $$\log(A+tB)=\log(A(I+tA^{-1}B)= \log A + \log(I+tA^{-1}B) = \log A + tA^{-1}B + \mathcal{O}(t^2)$$
  4. Sep 11, 2015 #3
  5. Sep 11, 2015 #4


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    This doesn't seem quite right, unless ## A ## and ## B ## commute.
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