(50g) Nth Fibonacci Number
02-26-2015, 10:45 PM
Post: #9
 Thomas Klemm Senior Member Posts: 1,608 Joined: Dec 2013
RE: (50g) Nth Fibonacci Number
This is the diagonalization of this matrix: When $$M$$ is squared the inner product of $$S^{-1} \cdot S$$ cancel:
$$M^2=(S \cdot J \cdot S^{-1})^2=S \cdot J \cdot S^{-1} \cdot S \cdot J \cdot S^{-1}=S \cdot J \cdot J \cdot S^{-1}=S \cdot J^2 \cdot S^{-1}$$
This can be generalized for all powers so that we end up with:
$$M^n=S \cdot J^n \cdot S^{-1}$$
But the power of a diagonalized matrix consists just of the power of its elements on the diagonal (i.e. the eigenvalues).
This brings us back to Eddie's initial solution. Okay, maybe after a little exercise in algebra.

Cheers
Thomas
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 Messages In This Thread (50g) Nth Fibonacci Number - Eddie W. Shore - 02-21-2015, 03:31 PM RE: (50g) Nth Fibonacci Number - Gerald H - 02-22-2015, 09:48 AM RE: (50g) Nth Fibonacci Number - Joe Horn - 02-22-2015, 09:06 PM RE: (50g) Nth Fibonacci Number - Offroad - 02-23-2015, 03:07 AM RE: (50g) Nth Fibonacci Number - rprosperi - 02-26-2015, 01:42 PM RE: (50g) Nth Fibonacci Number - Han - 02-26-2015, 07:39 PM RE: (50g) Nth Fibonacci Number - rprosperi - 02-26-2015, 08:23 PM RE: (50g) Nth Fibonacci Number - Joe Horn - 02-26-2015, 10:19 PM RE: (50g) Nth Fibonacci Number - Thomas Klemm - 02-26-2015 10:45 PM RE: (50g) Nth Fibonacci Number - Han - 02-27-2015, 03:29 AM RE: (50g) Nth Fibonacci Number - rprosperi - 02-27-2015, 01:31 PM RE: (50g) Nth Fibonacci Number - Thomas Klemm - 02-26-2015, 11:04 PM RE: (50g) Nth Fibonacci Number - Thomas Klemm - 02-27-2015, 05:10 AM RE: (50g) Nth Fibonacci Number - rprosperi - 02-27-2015, 01:43 PM RE: (50g) Nth Fibonacci Number - Thomas Klemm - 02-27-2015, 05:58 AM RE: (50g) Nth Fibonacci Number - rprosperi - 02-27-2015, 01:48 PM RE: (50g) Nth Fibonacci Number - Han - 02-27-2015, 02:22 PM RE: (50g) Nth Fibonacci Number - Gerald H - 02-27-2015, 03:27 PM RE: (50g) Nth Fibonacci Number - Thomas Klemm - 03-02-2015, 02:11 AM

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