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 Eigensystem of tetration-matrices bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 09/07/2007, 02:15 PM I think what you have done is similar what I describe now: Given the exponentiation to base s: $\exp_s$. We know that it has the (lower) fixed point t with $t^{1/t}=s$. Let $\tau_t(x)=x+t$ then $f:=\tau_t^{-1}\circ \exp_s \circ \tau_t$ is a function with fixed point 0 and with $f_1=f'(0)=s^{x+t}\ln(s)|_{x=0}=s^t\ln(s)=t\ln(s)=\ln(t)$. Further it is known that a power series f with $f_0=0$, $f_1\neq 1$ and $|f_1|\neq 1$ is conjugate to the linear function $\mu_{f_1}(x):=f_1x$. As in our case $1 and hence $1 we apply it to our $f$: $f=\alpha\circ \mu_{\ln(t)}\circ \alpha^{-1}$ $\exp_s = \tau_t\circ \alpha \circ \mu_{\ln(t)}\circ \alpha^{-1}\circ \tau_t^{-1}$ $\exp_s^{\circ x} = \tau_t\circ \alpha \circ \mu_{\ln(t)^x}\circ \alpha^{-1}\circ \tau_t^{-1}$. Now lets express this with power derivation matrices, let $P(f)$ be the power derivation matrix of $f$. Then we have: $P(\exp_s)=P(\tau_t)P(\alpha)P(\mu_{\ln(t)})P(\alpha)^{-1}P(\tau_t)^{-1}$. We have the following correspondences $P(\mu_a)={}_dV(a)$ $P(\exp)=B^{\sim}$ $P(\exp_s)=P(\exp\circ\mu_{\ln(s)})=P(\exp)P(\mu_{\ln(s)})=B^{\sim}{}_dV(\ln(s))=({}_dV(\ln(s))B)^{\sim}=B_s^{\sim}$ $P(\mu_{\ln(t)})={}_d\Lambda={}_dV(\ln(t))$ $P(\tau_t\circ\alpha)=(W_s^{-1})^{\sim}=(X_s P^{\sim} V(t))^{\sim}=V(t)^{\sim} P X_s^{\sim}$. Now lets have a look at the structure of $P(\tau_t)$. We can decompose $\tau_t = \mu_t\circ \tau_1\circ \mu_{\frac{1}{t}}$ where $P(\tau_1)$ is the lower triangular Pascal matrix given by $(\tau_1)_{m,n}=\left(m\\n\right)$. This is because the $m$th row of the power derivation matrix of $x+1$ consists of the coefficients of $(x+1)^m=\sum_{n=0}^m\left(m\\n\right)x^n$. As $P(\mu_t)=V(t)^{\sim}$ we conclude $P(\mu_{\frac{1}{t}}\circ \alpha)=X_s^{\sim}$. 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 $b>e^{1/e}$. « Next Oldest | Next Newest »

 Messages In This Thread Eigensystem of tetration-matrices - by Gottfried - 08/29/2007, 11:11 AM RE: Eigensystem of tetration-matrices - by Gottfried - 09/01/2007, 01:55 PM RE: Eigensystem of tetration-matrices - by Gottfried - 09/06/2007, 10:57 PM RE: Eigensystem of tetration-matrices - by Gottfried - 09/15/2007, 12:33 PM RE: Eigensystem of tetration-matrices - by Gottfried - 09/20/2007, 06:48 AM RE: Eigensystem of tetration-matrices - by Gottfried - 09/20/2007, 07:06 AM

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