The Fibonacci sequence by matrix similarity transform

(a) Since this method of calculating the Fibonacci sequence is rather round-about anyway, we can tolerate the inefficiency of the most obvious implementation:

import numpy as np

F = np.matrix('1 1; 1 0')

for n in range(10):
    print((F**n)[0,0], end=' ')
print()

(b) The necessary matrix eigenvalue, eigenvector and inverse operations are provided by numpy.linalg. Note that the value for $F_n$ obtained will be approximate for all but the smallest values of $n$ because of the finite floating point precision of the calculations.

import numpy as np

F = np.matrix('1 1; 1 0')
(lam1, lam2), C = np.linalg.eig(F)
Ci = np.linalg.inv(C)

def fib(n):
    D = np.matrix( [[lam1**n, 0], [0, lam2**n]] )
    M = C * D * Ci
    return int(M[0,0])

n = 1100
print('F({:d}) = {:d}'.format(n, fib(n-1)))