Talk:Euler-Binet Formula/Proof 2

This page discusses the contents of Euler-Binet Formula/Proof 2.

The reason why I put in the result of Eigenvalue of Matrix Powers was to make reasoning shorter: Without it, it would have been necessary to show that for square matrices $A$, $B$, and $D$ (where $D$ is invertible):
 * $DAD^\paren {-1} DBD^\paren {-1} = DABD^\paren {-1}$

Which itself is not hard to show. But then it would have been necessary to extend it to arbitrary many matrices $A_1, A_2, \ldots$. But even if not stated in this general form, I think it would not have been so easy to see the result after this "change of basis" applied to
 * $\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n$

Still, the Definition of nilpotent matrix has to be added.

Hope that this proof is a bit more insightful than merely proving by induction.

But at least it shows a way of deriving other closed-form solutions for similar sequences defined recursively -- the convergents of certain periodic simple continued fractions come to mind; or the Lucas numbers.


 * What you do is put the interim results into their own pages and then link to them. --prime mover (talk) 21:07, 11 January 2014 (UTC)


 * I hope that the main proof is now easier to follow. The current version is a compromise. Unfortunately I see no way how to put in further explanations. --Zahlenspieler (talk) 10:28, 15 January 2014 (UTC)


 * Still needs fixing in one or two places, someone will get round to that when they get round to it. Probably won't bother with the in-line reference tags, they seem not to be particularly reliable in this environment anyway. General tidying has been flagged up for action and will be done as and when. --prime mover (talk) 11:24, 15 January 2014 (UTC)