Gottfried Wrote: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.
Gottfried, we discussed that already. There is no diagonalization uniqueness for infinite matrices. For example the infinite Carleman/Bell Matrix of \( sqrt{2}^x \) can be diagonalized with eigenvalues being powers of \( \ln(2) \) and with eigenvalues being powers of \( \ln(4) \).
What we however have is a unique diagonalization (up to swapping eigenvalues of course) of the finite approximating matrices. This method is always applicable (especially also to functions with no fixed points, like \( e^x \)) and clearly defined.
Quote: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)
So the conjecture is that the eigenvalues of the truncated Carleman/Bell matrix of \( b^x \) converge to the set of powers of \( \ln(a) \) where \( a \) is the lower (the attracting) fixed point of \( b \)?
Quote: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"
Yes, as I demonstrated in "diagonal vs natural" it makes absolutly sense to consider \( E^t \) where \( E \) is the Carleman/Bell-Matrix of \( e^x \). Even if the Eigenvalues behave strangely (dont converge to powers of anything) you have for each truncated matrix a unique real \( t \)-th power (how powers or generally holomorphic functions are applied to matrices by diagonalization is explained here) and the values on the first row/column of this power converge with increasing matrix size.