Matrix functions

  • #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.
 
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Answers and Replies

  • #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:)
 
  • #3
pwsnafu
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Offtopic...

(btw, i don't bother to write matrices in bold: i write vectors bold, scalars small plain, and matrices big plain :wink:)

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:
 
  • #4
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:)

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]?
 
  • #5
tiny-tim
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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]?

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
 
  • #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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