Matrix-method: compare use of different fixpoints

[Gottfried]

I simply dont see how you get fixed points involved.
You have Carleman matrix \( B_b \) of \( b^x \) and you uniquely decompose the finite truncations \( {B_b}_{|n} \) into
\( {B_b}_{|n}=W_{|n} D_{|n} {W_{|n}}^{-1} \) and then you define

\( {B_b}^t = \lim_{n\to\infty} W_{|n} {D_{|n}}^t {W_{|n}}^{-1} \).

Where are the fixed points used?

Assume the eigensystem-decomposition Bs = W0 * D0 * W0^-1
where the small s indicates the baseparameter s=b=base

Now this means also
W0^-1 * Bs = D0 * W0^-1

Now look at the first row of W0^-1, call this vector Y0. Then, since I assume the first eigenvalue d0_0 =1 we have
Y0 * Bs = 1 * Y0 = Y0
Now I observed, that Y0 has the form of a powerseries of the scalar parameter y0, so I may note,

V(y0)~ * Bs = V(y0)~

But on the other hand I know by construction of Bs that
V(x)~ * Bs = V(s^x)~

so the previous means
V(y0)~ * Bs = V(y0)~ = V(s^y0) ~
thus
y0 = s^y0

and y0 is obviously a fixpoint.

The observation actually was: the first row in W^-1 is the powerseries in the first fixpoint y0.

Now consider this backways. Using another fixpoint y1 this relation holds again,
y1 = s^y1
V(y1)~ * Bs = V(s^y1)~ = V(y1)~
First row of W1^-1 is V(y1)~
and W0^-1 <> W1^-1 in their first rows and supposedly the same for the whole matrices W0 <> W1 and

W1^-1 * Bs = 1 * W1^-1

is another solution, provided that the form of the first row in W1^-1 is still that of a powerseries.

And in fact; if I introduce the coefficients u1 = alpha + beta*I and t1=exp(u1) in my analytical eigensystem-constructor as parameters, I get valid eigen-matrices W1 and W1^-1 based on these other fixpoints for the cases I've checked where it holds that
Bs = W1*D1*W1^-1 (but see note 1)

These had in fact the powerseries of the second fixpoint in the first row in W1^-1 - and I suppose, this will be the same with all other fixpoints (however the numerical computations become too difficult to do a general conjecture based on and backed by reliable approximations)

Gottfried
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(1) This is directly related to your introductory posting in the "Bummer"-thread, and we must solve the discrepancy between these two statements!