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Matrix functions

  1. Mar 15, 2013 #1
    I'm trying to expand a function of a bilinear map in a power series [itex]f({\bf x^{T}Ax})[/itex]. It isn't quite a matrix function because it takes a matrix and a vector and maps them into a scalar. I'd like to expand it into powers of [itex]\bf A[/itex], but still preserve the function as a scalar. As far as I can tell, a matrix power series takes a matrix as an input (like this one) but outputs a matrix (unlike this problem). I'm not sure how to proceed.
     
    Last edited: Mar 15, 2013
  2. jcsd
  3. Mar 16, 2013 #2

    tiny-tim

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    hi unchained1978! :smile:

    if A = ∑ Bn, then xTAx = ∑ xTBnx :wink:

    (btw, i don't bother to write matrices in bold: i write vectors bold, scalars small plain, and matrices big plain :wink:)
     
  4. Mar 16, 2013 #3

    pwsnafu

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    Offtopic...

    I use the convention from linear analysis: scalars are lower case Greek, vectors are lower case Latin, and operators are upper case Latin. No need to worry about boldface :wink:
     
  5. Mar 16, 2013 #4
    I'm not sure I understand how that relates to my problem. Lets say I have a function [itex]f({\bf x^{T}}A{\bf x})=\frac{1}{{\bf x^{T}}A{\bf x}}[/itex]. Are you saying this is the same as [itex]{\bf x^{T}}(1+A^{2}+A^{3}+...+A^{n}){\bf x}[/itex]?
     
  6. Mar 16, 2013 #5

    tiny-tim

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    hmm … i didn't realise that's what you meant :redface:

    let's use 1/(xT(1 - A)x) instead …

    that's 1/(xT(1/∑An)x) …

    no, i don't see how you can ever get those xs from the bottom to the top
     
  7. Mar 16, 2013 #6
    Oops. I forgot to write 1/(x^T (1-A)x). So what's the general procedure for this sort of problem? I'm working with a Gaussian function [itex]e^{-{\bf x^{T}}A{\bf x}}[/itex] and I'd like to expand that in A? Any suggestions?
     
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