diagonal vs regular

[Gottfried]

Hmm, the fixpoints-shifts resulted in similarity scalings with the matrix-method (using powers of the pascalmatrix). Then the eigenvalues should be unchanged.

I dont know whether they should, but actually there are different. I would say that this has to do with that you need an infinite matrix to do a fixed point shift. While you are actually dealing with a finite matrix.

... are different, if truncated? If so, ok.
My conversion/factorizing of the infinite square-matrix into a triangular one by similarity scaling (using powers of the pascal-matrix) applies to the infinite case. I'd like to see, how the eigenvalues for the infinite square-matrix are determined otherwise.
When I started my tetration-discussion based on diagonalization, I considered the sequences of sets of eigenvalues for truncated matrices with increasing size, where the parameter b for f:=b^x was in the "range of convergence", so approximations to the limit-case for the sets of eigenvalues made sense. See for instance sets of eigenvalues b=1.7^(1/1.7)
The approximation to a sequence of powers of a constant (here of log(1.7)) was the reason for my hypothesis, that this is in principle also true for the parameters b out of the "range of convergence"