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Function of operators. Series, matrices.

  1. May 26, 2014 #1
    Is it possible to write series ##\ln x=\sum_na_nx^n##. I am asking you this because this is Taylor series around zero and ##\ln 0## is not defined.

    And if ##A## is matrix is it possible to write
    ##\ln A=\sum_na_nA^n##. Thanks for the answer!
     
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  3. May 26, 2014 #2

    mathman

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    The answer to the first question is no. This forces the second question to also have an answer no.

    For ln if you want a series you can use ln(1+x) or ln(1-x) or ln((1+x)/(1-x)).
     
  4. May 26, 2014 #3
    Ok. Tnx. But it is ok to write
    ##lnx=\sum_n a_n(x-1)^n ##
    right?
    or
    ##lnA=\sum_n a_n(A-1)^n ##
     
  5. May 26, 2014 #4

    HallsofIvy

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    Yes, of course. In fact it is well known that (Taylor's series)
    [tex]ln(x)= \sum_{n=1}^\infty \frac{(-1)^n}{n}(x- 1)^n[/tex]
     
  6. May 27, 2014 #5
    Ok. Tnx. Is it true also for operators?
     
  7. May 27, 2014 #6

    micromass

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    Yes, but not for all operators. Only the ones in a certain radius of convergence. Here, that would be that it's ok for all operators with ##\|A\|<1##.

    For a proof, see "the structure and Geometry of Lie groups" by Hilgert and Neeb, chapter 3. Although most functional analysis texts should have this as well.
     
  8. May 28, 2014 #7
    Thanks a lot. My problem is to calculate ##Tr(A\ln A)##, where ##A## is matrix. Because of that I thought to write ##\ln A## as power series if that is possilble. But here I can not do it in such easy way! Right?
     
  9. May 28, 2014 #8

    micromass

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    If ##\|A\|<1##, then you can do it this way. Otherwise, you will need to tell us what your definition of ##\ln A## is.
     
  10. May 28, 2014 #9
    Is A by any chance a density matrix (ie are you calculating the von Neumann entropy)? If so, its properties guarantee that you can use its spectral decomposition to write the trace as [itex]\mathrm{Tr} \rho \ln \rho = \sum_i \lambda_i \ln{\lambda_i}[/itex] for the eigenvalues [itex]\lambda_i[/itex] of the density matrix, which is often a useful form.
     
  11. May 28, 2014 #10
    When can I use spectral decomposition? Is it true only for Hermitian matrices, or not? Also what will happen in case of degenerated spectrum?
     
  12. May 28, 2014 #11
    It exists for all normal operators, so any Hermitian matrices are fine. For a degenerate spectrum, you can use the general spectral decomposition: [itex]A=\sum_{i=1}^{n}\sum_{d=1}^{D(i)} \lambda_i |\lambda_i,d\rangle\langle\lambda_i,d|[/itex], where D(i) is the degeneracy of eigenvalue [itex]\lambda_i[/itex], so that the vectors [itex]{|\lambda_i,d\rangle}[/itex] form an orthonormal basis on the corresponding D(i)-dimensional eigenspace.

    Note that other properties of the density matrix would guarantee that the logarithm is always well-defined: The operator norm of a bounded normal operator equals its spectral radius, and if this is indeed a density operator, its matrix elements are non-negative and [itex]\sum_i \lambda_i = 1[/itex]. So, the condition micromass mentioned, ||A||<1, is fulfilled, unless the density matrix describes a pure state, that is, one of the eigenvalues of A equals one, in which case the von Neumann entropy vanishes.
     
  13. May 28, 2014 #12
    In case of degeneracy I can not find eigenvectors in unique form. So I think that I can not find operator in unique form. For example look the case in ##\mathbb{R}^3##, ##\lambda_1=\lambda_2=\lambda_3=1## and eigenvectors
    ##x_1^T=(1 \quad 0 \quad 0)##,##x_2^T=(0 \quad 0 \quad 0)##, ##x_3^T=(0 \quad 0 \quad 0)##. How to find matrix in this case?
     
  14. May 28, 2014 #13
    Eigenvectors cannot be zero vectors.

    (In addition, if we're still considering density matrices, that's not going to be a valid density matrix since the sum of eigenvalues is greater than 1.)
     
  15. May 28, 2014 #14

    WWGD

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  16. Jun 3, 2014 #15

    HallsofIvy

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    Yes, most texts specify that eigenvectors must be non-zero. But that leaves us having to say "the set of all eigenvectors of a linear transformation, together with the zero vector, form a sub-space" and "the set of all eigenvectors corresponding to a given eigenvector, together with the zero vector, form a subspace."

    A few texts say "[itex]\lambda[/itex] is an eigenvalue of linear transformation A if and only if there is a non-zero vector, v, such that [itex]Av= \lambda v[/itex]" and then "v is an eigenvector of A, corresponding to eigenvalue [itex]\lambda[/itex], if and only if [itex]Av= \lambda v[/itex]" which does NOT require that an eigenvector be non-zero. That allows us to say simply "the set of all eigenvectors of a linear transformation form a subspace" without having to add the zero vector. A small point but I see no reason to refuse to allow the zero vector as an eigenvector.
     
  17. Jun 5, 2014 #16
    Really? Mind giving a reference? I'm not realy skeptical, just curious, since I have never seen that convention being used in a text (either in linear algebra or in functional analysis), and this is the second time someone commented about that on a post I made. I guess the main difference is that you give up the uniqueness of the eigenvalue of a set eigenvector, although I wouldn't be surprised if more exceptions arise.
     
    Last edited: Jun 5, 2014
  18. Jun 5, 2014 #17

    micromass

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    I have never seen a math text that allows zero as an eigenvector. I would like to see some references for this too.
     
  19. Jun 5, 2014 #18

    pasmith

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    In my usage, an eigenvector corresponding to an eigenvalue [itex]\lambda[/itex] is a non-zero [itex]v \in V[/itex] such that [itex]Av = \lambda v[/itex]. The eigenspace corresponding to the eigenvalue [itex]\lambda[/itex] is the set of all [itex]v \in V[/itex] such that [itex]Av = \lambda v[/itex]; the eigenspace is, as the name suggests, a subspace of [itex]V[/itex].
     
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