Fibonacci Linear Algebra Matrix Formulation Plus

 Just the key matrices will be presented here.

Above: Formula is set out with conditions stated

  

 
 
 Following that, using the rules of linear algebra, we get the basic matrix equation as shown above.
 

Finally, we get the Fibonacci equation in the most compact matrix form as shown above.. Wooohooooo!
 
 
The following are extras depending on whether one needs elucidation or not:
 

 Further elucidation is shown above
 
 
 
The graphic above shows an application of the Fibonacci sequence algorithm intended for Python computer programming language. 
* means matrix multiplication
 
 

Diagonalization and Fibonacci Sequence: Watch - https://www.youtube.com/watch?v=Atrsn5OriIw


Fibonacci Numbers and Spirals - https://sites.google.com/a/icsd.k12.ny.us/the-golden-ratio-2015-2016/home/fibonacci-numbers-and-spirals


Fibonacci and the Golden Ratio: Divine Geometry?: https://www.bbvaopenmind.com/en/science/mathematics/fibonacci-and-the-golden-ratio-divine-geometry/


Research and Reflection: Fractals, the Fibonacci Spiral, and Nature - https://blogs.uoregon.edu/mjanesaad199/scientific-research-fractals-the-fibonacci-spiral-and-nature/

 

How to draw the Fibonacci Spiral - Illustration involving the first few inputs..


 




 

 

 

 



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What is the coefficient matrix of the Fibonacci sequence equation?

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        The Fibonacci sequence is a sequence of numbers in which each number is the sum of the two preceding ones, starting from 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, 21, ...   The Fibonacci sequence can be represented by the linear homogeneous recurrence relation:  F(n) = F(n-1) + F(n-2)  Where F(n) represents the n-th number in the sequence.  To find the coefficient matrix for this equation, we can write it in matrix form as:  [F(n), F(n-1)] = [F(n-1), F(n-2)] * [1, 1; 1, 0]  The matrix [1, 1; 1, 0] is the coefficient matrix for this equation.  It represents the transformation that maps the n-1th and n-2nd numbers in the sequence to the nth number.  This matrix is called the Fibonacci matrix and it plays a crucial role in various areas of mathematics and science, including number theory, graph theory, and the study of dynamical systems.
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