I have a circulant matrix defined by a positive kernel W(x):
$W_{ij} = W(|i-j|)$
where W is defined on positive reals (so we are sampling {1,2,...,N} to create the matrix). I know this has eigenvectors which are the Fourier basis and constructed using the Nth roots of unity.
Now, if W(x) depends on a parameter r that varies very slowly with the row of the matrix:
W = W(r,x)
so that the new matrix is now defined by:
$W_{ij} = W(r(i),|i-j|) $
How do I find the new eigenvectors of this matrix?
Numerically I see that they are localized.