Getting eigenvalues from 3x3 matrix in Python using Power method

问题

I'm trying to get all eigenvalues from a 3x3 matrix by using Power Method in Python. However my method returns diffrent eigenvalues from the correct ones for some reason.

My matrix: A = [[1, 2, 3], [2, 4, 5], [3, 5,-1]]

Correct eigenvalues: [ 8.54851285, -4.57408723, 0.02557437 ]

Eigenvalues returned by my method: [ 8.5485128481521926, 4.5740872291939381, 9.148174458392436 ]

So the first one is correct, second one has wrong sign and the third one is all wrong. I don't know what I'm doing wrong and I can't see where have I made mistake.

Here's my code:

import numpy as np
import numpy.linalg as la

eps = 1e-8 # Precision of eigenvalue

def trans(v): # translates vector (v^T)
    v_1 = np.copy(v)
    return v_1.reshape((-1, 1))

def power(A):
    eig = []
    Ac = np.copy(A)
    lamb = 0
    for i in range(3):
        x = np.array([1, 1, 1])
        while True:
            x_1 = Ac.dot(x) # y_n = A*x_(n-1)
            x_norm = la.norm(x_1) 
            x_1 = x_1/x_norm # x_n = y_n/||y_n||
            if(abs(lamb - x_norm) <= eps): # If precision is reached, it returns eigenvalue
                break
            else:
                lamb = x_norm
                x = x_1
        eig.append(lamb)

        # Matrix Deflaction: A - Lambda * norm[V]*norm[V]^T
        v = x_1/la.norm(x_1)
        R = v * trans(v)
        R = eig[i]*R
        Ac = Ac - R

    return eig

def main():
    A = np.array([1, 2, 3, 2, 4, 5, 3, 5, -1]).reshape((3, 3))
    print(power(A))



if __name__ == '__main__':
    main()

PS. Is there a simpler way to get the second and third eigenvalue from power method instead of matrix deflaction?

回答1:

With

lamb = x_norm

you ever only compute the absolute value of the eigenvalues. Better compute them as

lamb = dot(x,x_1)

where x is assumed to be normalized.

As you do not remove the negative eigenvalue -4.57408723, but effectively add it instead, the largest eigenvalue in the third stage is 2*-4.574.. = -9.148.. where you again computed the absolute value.

回答2:

I didn't know this method, so I googled it and found here:

http://ergodic.ugr.es/cphys/LECCIONES/FORTRAN/power_method.pdf

that it is valid only for finding the leading (largest) eigenvalue, thus, it seems that it is working for you fine, and it is not guaranteed that the following eigenvalues will be correct. Btw. numpy.linalg.eig() works faster than your code for this matrix, but I am guessing you implemented it as an exercise.

来源：https://stackoverflow.com/questions/48385630/getting-eigenvalues-from-3x3-matrix-in-python-using-power-method

标签

python

numpy

linear-algebra

numerical-methods