Expanding Bilinear Map Power Series with Matrix/Vector Input

[unchained1978]

I'm trying to expand a function of a bilinear map in a power series $f({\bf x^{T}Ax})$. 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 $\bf A$, 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.

 

[tiny-tim]

hi unchained1978!

if A = ∑ Bⁿ, then x^TAx = ∑ x^TBⁿx

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

 

[pwsnafu]

Offtopic...

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

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

 

[unchained1978]

hi unchained1978!

if A = ∑ Bⁿ, then x^TAx = ∑ x^TBⁿx

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

I'm not sure I understand how that relates to my problem. Let's say I have a function $f({\bf x^{T}}A{\bf x})=\frac{1}{{\bf x^{T}}A{\bf x}}$. Are you saying this is the same as ${\bf x^{T}}(1+A^{2}+A^{3}+...+A^{n}){\bf x}$?

 

[tiny-tim]

I'm not sure I understand how that relates to my problem. Let's say I have a function $f({\bf x^{T}}A{\bf x})=\frac{1}{{\bf x^{T}}A{\bf x}}$. Are you saying this is the same as ${\bf x^{T}}(1+A^{2}+A^{3}+...+A^{n}){\bf x}$?

hmm … i didn't realize that's what you meant

let's use 1/(x^T(1 - A)x) instead …

that's 1/(x^T(1/∑Aⁿ)x) …

no, i don't see how you can ever get those xs from the bottom to the top

 

[unchained1978]

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 $e^{-{\bf x^{T}}A{\bf x}}$ and I'd like to expand that in A? Any suggestions?