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Eigensystem of tetration-matrices
#4
I think what you have done is similar what I describe now:

Given the exponentiation to base s: .
We know that it has the (lower) fixed point t with .
Let then

is a function with fixed point 0 and with .

Further it is known that a power series f with , and is conjugate to the linear function . As in our case and hence we apply it to our :


.

Now lets express this with power derivation matrices, let be the power derivation matrix of . Then we have:
.

We have the following correspondences




.

Now lets have a look at the structure of . We can decompose where is the lower triangular Pascal matrix given by . This is because the th row of the power derivation matrix of consists of the coefficients of . As we conclude
.

I think this completes the correspondences.
Unfortunately this decomposition works merely if the function under consideration has a fixed point. In so far it is very interesting that it also converges for the non-fixed point case with base .
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Messages In This Thread
Eigensystem of tetration-matrices - by Gottfried - 08/29/2007, 11:11 AM

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