EIgenvalues & Eigenvectors.ipynb

Kernel: Python 3 (system-wide)

Matrices & Vectors

A=np.array([[1,2],[3,4]])
v=np.array([-1,2])
print(A)
print("v =",v)
print("Av = ",np.dot(A,v))

[[1 2]
 [3 4]]
v = [-1  2]
Av =  [3 5]

Power iteration

import numpy as np

np.set_printoptions(precision=3)

# power iteration / von Mises iteration
# gives an approximation for an eigenvector for the dominant eigenvalue of a matrix A
# by doing the power iteration k times
def power_iteration(A, k: int):
    v = np.random.rand(A.shape[1])

    for _ in range(k):
        # calculates b = Av
        b = np.dot(A, v)

        # calculate the norm of b
        b_norm = np.linalg.norm(b)

        # define v to be the normalized version of b
        v = b / b_norm

    return v

def give_eigenvalue(A,k):
    v=power_iteration(A,k)
    lam = np.dot(A,v)[0]/v[0]
    
    return lam

# Using power iteration example
A=np.array([[1,2],[3,4]])
print(A)

v=power_iteration(A,3)

print(v)
print(np.dot(A,v))
print(np.dot(A,v)[0]/v[0])
print(np.dot(A,v)[1]/v[1])

give_eigenvalue(A,10)

[[1 2]
 [3 4]]
[0.416 0.909]
[2.235 4.885]
5.371605514503671
5.37249346495099

5.372281323264108

print("Eigenvector from power-it:", power_iteration(A,1))
print("Eigenvalue from power-it:", give_eigenvalue(A,1))

print("Eigenvector from power-it:", power_iteration(A,5))
print("Eigenvalue from power-it:", give_eigenvalue(A,5))

print("Eigenvector from power-it:", power_iteration(A,30))
print("Eigenvalue from power-it:", give_eigenvalue(A,30))

Eigenvector from power-it: [0.35 0.93]
Eigenvalue from power-it: 5.193779822316735
Eigenvector from power-it: [0.42 0.91]
Eigenvalue from power-it: 5.372280232439236
Eigenvector from power-it: [0.42 0.91]
Eigenvalue from power-it: 5.372281323269014

# Eigenvectors and eigenvalues are implemented in numpy as follows:
print("Eigenvectors:")
print(np.linalg.eig(A))
print("Eigenvalues: ")
print(np.linalg.eigvals(A))

Eigenvectors:
(array([-0.37,  5.37]), array([[-0.82, -0.42],
       [ 0.57, -0.91]]))
Eigenvalues: 
[-0.37  5.37]

---

(1) Population behavior / Leslie matrices

A=np.array([[0,3,2],[4/5,0,0],[0,2/5,0]])
print(A)
print("Dominant eigenvalye:", give_eigenvalue(A,100))
print("Eigenvector:", 100*power_iteration(A,100))

[[0.  3.  2. ]
 [0.8 0.  0. ]
 [0.  0.4 0. ]]
Dominant eigenvalye: 1.6684119203677292
Eigenvector: [89.69 43.01 10.31]

---

(2) Google Page rank

A=np.array([[0,1/3,0,0,0,0],[1,0,1/2,0,0,0],[0,1/3,0,1/2,0,1/2],[0,1/3,1/2,0,1/2,1/2],[0,0,0,1/2,0,0],[0,0,0,0,1/2,0]])
print(A)
print("Dominant eigenvalye:", give_eigenvalue(A,100))
print("Eigenvector:", 1/0.16*power_iteration(A,100))

[[0.   0.33 0.   0.   0.   0.  ]
 [1.   0.   0.5  0.   0.   0.  ]
 [0.   0.33 0.   0.5  0.   0.5 ]
 [0.   0.33 0.5  0.   0.5  0.5 ]
 [0.   0.   0.   0.5  0.   0.  ]
 [0.   0.   0.   0.   0.5  0.  ]]
Dominant eigenvalye: 1.0
Eigenvector: [0.83 2.5  3.33 4.   2.   1.  ]

---

(3) Decision making

A=np.array([[1,4,2,3/2],[1/4,1,1/2,1/3],[1/2,2,1,1],[2/3,3,1,1]])
print(A)
print("Dominant eigenvalye:", give_eigenvalue(A,1000))
print("Eigenvector:", power_iteration(A,1000))

[[1.   4.   2.   1.5 ]
 [0.25 1.   0.5  0.33]
 [0.5  2.   1.   1.  ]
 [0.67 3.   1.   1.  ]]
Dominant eigenvalye: 4.016358973921448
Eigenvector: [0.75 0.18 0.41 0.48]