Numpy - Modal matrix and diagonal Eigenvalues

Question

I wrote a simple Linear Algebra code in Python Numpy to calculate the Diagonal of EigenValues by calculating $M^{-1}.A.M$ (M is the Modal Matrix) and it's working strange.

Here's the Code :

import numpy as np

array = np.arange(16)
array = array.reshape(4, -1)
print(array)

[[ 0  1  2  3]
 [ 4  5  6  7]
 [ 8  9 10 11]
 [12 13 14 15]]

eigenvalues, eigenvectors = np.linalg.eig(array)

print eigenvalues
[  3.24642492e+01  -2.46424920e+00   1.92979794e-15  -4.09576009e-16]

print eigenvectors
[[-0.11417645 -0.7327781   0.54500164  0.00135151]
 [-0.3300046  -0.28974835 -0.68602671  0.40644504]
 [-0.54583275  0.15328139 -0.2629515  -0.8169446 ]
 [-0.76166089  0.59631113  0.40397657  0.40914805]]

inverseEigenVectors = np.linalg.inv(eigenvectors) #M^(-1)
diagonal= inverseEigenVectors.dot(array).dot(eigenvectors) #M^(-1).A.M

print(diagonal)
[[  3.24642492e+01  -1.06581410e-14   5.32907052e-15   0.00000000e+00]
 [  7.54951657e-15  -2.46424920e+00  -1.72084569e-15  -2.22044605e-16]
 [ -2.80737213e-15   1.46768503e-15   2.33547852e-16   7.25592561e-16]
 [ -6.22319863e-15  -9.69656080e-16  -1.38050658e-30   1.97215226e-31]]

the final 'diagonal' matrix should be a diagonal matrix with EigenValues on the main diagonal and zeros elsewhere. but it's not... the two first main diagonal values ARE eigenvalues but the two second aren't (although just like the two second eigenvalues, they are nearly zero).

and by the way a number like $-1.06581410e-14$ is literally zero so how can I make numpy show them as zero?

What am I doing wrong?

Thanks...

Answer 1

Just round the final result to the desired digits :

print(diagonal.round(5))

array([[ 32.46425,   0.     ,   0.     ,   0.     ],
       [  0.     ,  -2.46425,   0.     ,   0.     ],
       [  0.     ,   0.     ,   0.     ,   0.     ],
       [  0.     ,   0.     ,   0.     ,   0.     ]])

Don't confuse precision of computation and printing policies.

Answer 2

>>> diagonal[np.abs(diagonal)<0.0000000001]=0
>>> print diagonal
[[ 32.4642492   0.          0.          0.       ]
 [  0.         -2.4642492   0.          0.       ]
 [  0.          0.          0.          0.       ]
 [  0.          0.          0.          0.       ]]
>>>