Talk:Euler-Binet Formula/Proof 2

This page deals with 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):
 * $\displaystyle DAD^\left({-1}\right)DBD^\left({-1}\right) =DABD^\left({-1}\right)$

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
 * $\displaystyle \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n$

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

Hope that this proof is a bit more insightful than merely prving 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)