Discussion Overview
The discussion revolves around expanding a function of a bilinear map in a power series, specifically focusing on the expression f({\bf x^{T}Ax}). Participants explore the challenges of maintaining the scalar output while manipulating matrix inputs, and they consider various forms of power series expansions.
Discussion Character
- Exploratory, Technical explanation, Debate/contested
Main Points Raised
- One participant expresses uncertainty about how to expand the bilinear map function f({\bf x^{T}Ax}) into powers of A while preserving its scalar nature.
- Another participant suggests that if A can be expressed as a sum of matrices Bn, then the expression xTAx can also be represented as a sum involving those matrices.
- A participant questions the relevance of the previous suggestion to their specific problem involving the function f({\bf x^{T}}A{\bf x}) = \frac{1}{{\bf x^{T}}A{\bf x}} and seeks clarification on whether it can be expressed as a power series.
- There is a proposal to use a different function, 1/(xT(1 - A)x), as a potential alternative for expansion, but uncertainty remains about the feasibility of manipulating the terms correctly.
- A participant introduces a Gaussian function e^{-{\bf x^{T}}A{\bf x}} and inquires about the general procedure for expanding this function in terms of A.
Areas of Agreement / Disagreement
Participants do not appear to reach a consensus on the methods for expanding the bilinear map or the Gaussian function, with multiple competing views and uncertainties present throughout the discussion.
Contextual Notes
Participants express varying conventions for notation, which may affect clarity in communication. There are unresolved questions regarding the manipulation of terms in the proposed expansions.
