Fibonacci Numbers and the Golden Ratio - An Eigenvalue problem
October 13, 2009
Fibonacci Numbers and the Golden Ratio are used quite extensively by technical analysts - men and women who study and analyze historical charts of stocks, currencies and other financial assets - to predict stock (currency) market moves. However, even though both the Fibonacci numbers and the Golden Ratio are staple diet for technical analysts in analyzing stock and currency charts, many would have trouble grasping the relationship between these two mathematical constructs.
What is the relationship between Fibonacci numbers and the Golden Ratio? As it turns out Golden Ratio is one of the eigenvalues of a Fibonacci matrix.
A Fibonacci sequence of numbers is given by:
0, 1, 1, 2, 3, 5, 8, 13, 21, ...........
Where, each number, except for zero and one, is the sum of the previous two numbers. In general they can be written as, with seed values, and :
Golden Ratio is the irrational number 1.6180339....... It is generally denoted by the Greek alphabet (psi) and is an extremely important ratio that is found in mathematics, nature and the world of arts. Some mathematicians, represent it as: . And interestingly, many Renaissance artists believed that the Golden Ratio was a "divine proportion".
We can write a Fibonacci sequence (of two successive Fibonacci numbers) as a 2 x 1 column vector such as:
As can be seen the above equation represents an eigenvalue problem where is the Fibonacci matrix.
It can be easily shown that one of the eigenvalues, of is equal to the Golden Ratio, 1.6180339....
Setting up the characteristic equation as where, is the eigenvalue of we get the quadratic equation:
Solving the above, we get: where:
Hence, one of the eigenvalues of A, is equal to the irrational number 1.6180339... which is the Golden Ratio. This is a really amazing result.
*Thanks to Prof. Megumi Harada at the McMaster University, Ontario, Canada.
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