I'm trying to understand the behavior the Singular Value Decomposition on a deeper level, and why it might give a particular result. Take the function
$$ f(x,y) = \frac{1}{(1+2x+y)^2} $$
and consider a continous range of parameters over $x \in [0,1]$, $y \in [0,.5]$. Consider an "infinite" matrix
$$ A = \begin{bmatrix} f(0,0) & f(\delta_1, 0) & f(2\delta_1,0) & ... & f(1,0) \\ f(\delta_1,\delta_2) & f(\delta_1, \delta_2) & f(2\delta_1, \delta_2) & ... & f(1,\delta_2) \\ ... & ... & ... & ... & ... \\ f(\delta_1,.5) & f(\delta_1, .5) & f(2\delta_1, .5) & ... & f(1,.5) \\ \end{bmatrix} $$
where $\delta_1$ and $\delta_2$ can be made as small as I like. I can certainly take a finite, but small value for the $\delta's$ and compute the SVD
$$A=USV$$
This gives some interesting behavior (plotted below in clockwise order $f$, log of the singular values, and the corresponding vectors, $u_i$ and $v_i$):
Questions:
Terminology: Is it correct to think of $f$ as a (normal) operator, or is it $A$? What is this operator?
Spectral theory: How do you compute the eigenvectors directly from $f$?