Difference in eigenvector transformations: Mathematica vs. SciPy

问题

Similar questions have been asked previously here but none seem to answer my example. I compute the eigenvalues and eigenvectors of a matrix A using Mathematica and SciPy; the eigenvalues agree but this is not the case for the eigenvectors:

(1) the lowest (eigenvalued) eigenvector agrees

(2) the remaining corresponding eigenvectors of Mathematica and SciPy are not related by a multiplicative factor

(3) I can compute the transformation matrix T sending SciPy's eigenvector to Mathematica's corresponding eigenvector using the outer product

T = numpy.outer(MathematicaEigenvector, SciPyEigenvector)

such that

MathematicaEigenvector = numpy.dot(T, SciPyEigenvector)

I would expect that the transformation matrix T should be the same for all SciPy-Mathematica eigenvector pairs because T is simply the matrix relating the eigenvectors of the matrix inv(T).A.T to that of the original matrix A. However performing step (2) for each of the eigenvector pairs gives different T matrices.

Can somebody explain this? I can post the matrices if required.

UPDATE: The python code and matrices are as follows:

S = [[0., -1, -1, -1, 0, 0, -1, 0, 0], 
    [-1, 0., -1, 0, -1, 0, 0, -1, 0], 
    [-1, -1, 0., 0, 0, -1, 0, 0, -1], 
    [-1, 0, 0, 0., -1, -1, -1, 0, 0], 
    [0, -1, 0, -1, 0., -1, 0, -1, 0], 
    [0, 0, -1, -1, -1, 0, 0, 0, -1], 
    [-1, 0, 0, -1, 0, 0, 0., -1, -1], 
    [0, -1, 0, 0, -1, 0, -1, 0., -1], 
    [0, 0, -1, 0, 0, -1, -1, -1, 0.]];

eig_val,eig_vec = scipy.linalg.eig(S)
idx = eig_val.argsort()
eig_val = np.array(eig_val[idx])
eig_vec = np.array(eig_vec[:,idx])

The Mathematica eigenvectors are:

[-0.333333, -0.333333, -0.333333, -0.333333, -0.333333, -0.333333, -0.333333, -0.333333, -0.333333], 
[0.0385464, 0.570914,   0.371276, -0.570914, -0.0385464, -0.238184, -0.33273, 0.199638,   0.], 
[0.570246, -0.0269007, 0.197029,   0.0269007, -0.570246, -0.346316, 0.373217, -0.22393,   0.], 
[-0.0816497, 0.0816497, -0.489898, -0.0816497,   0.0816497, -0.489898, 0.408248, 0.571548,   0.], 
[-0.333333, -0.333333, 0.166667, -0.333333, -0.333333,   0.166667, 0.166667, 0.166667, 0.666667], 
[-0.288675, 0.288675,   2.498e-16, -0.288675, 0.288675, -1.94289e-16,   0.57735, -0.57735, 0.],
[-0.5, 0.5, -2.04678e-16, 0.5, -0.5,   2.41686e-16, -9.25186e-17, 5.55112e-17, 0.], 
[0.166667,   0.166667, -0.333333, 0.166667,   0.166667, -0.333333, -0.333333, -0.333333, 0.666667], 
[0.288675,   0.288675, -0.57735, -0.288675, -0.288675, 0.57735,   4.02456e-16, -2.08167e-16, 0.]

Whereas the SciPy eigenvectors are:

[-0.33333333 -0.33333333 -0.33333333 -0.33333333 -0.33333333 -0.33333333 -0.33333333 -0.33333333 -0.33333333]
[ 0.12054181 -0.17813781  0.50013951  0.08577902 -0.21290061  0.4653767 -0.2872389  -0.58591853  0.0923588 ]
[ 0.12191583 -0.21327897  0.26215377 -0.28683603 -0.62203084 -0.1465981 0.35987707  0.02468226  0.500115  ]
[ 0.66666667  0.16666667  0.16666667  0.16666667 -0.33333333 -0.33333333 0.16666667 -0.33333333 -0.33333333]
[-0.16604424 -0.59504716 -0.43689399  0.43294845  0.00394553  0.16209871 0.43294845  0.00394553  0.16209871]
[-0.01305419  0.07446538 -0.0614112  -0.54881726  0.36347168  0.18534558 0.56187145 -0.43793706 -0.12393438]
[-0.66666667  0.33333333  0.33333333  0.33333333 -0.16666667 -0.16666667 0.33333333 -0.16666667 -0.16666667]
[-0.21052033  0.65306873 -0.4425484   0.10526016 -0.32653437  0.2212742 0.10526016 -0.32653437  0.2212742 ]
[-0.02303417  0.0714558  -0.04842162  0.09679298  0.41311466 -0.50990763 -0.0737588  -0.48457045  0.55832926]
[ 4.67737437  0.12612917  0.75157798 -0.09378424  0.91674876  2.36234989 1.03706802 -9.0725069   0.        ]

Both the above are ordered by the eigenvalues [-4.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, 2.+0.j, 2.+0.j, 2.+0.j, 2.+0.j]

回答1:

I believe the reason is as follows: because there are repeated eigenvalues the transformation matrix T must act on a linear combination of the eigenvectors in that subspace as opposed to individual eigenvalues. That is, my first code snippet should be modified to:

T = numpy.outer(MathematicaEigenvectorSubspace, SciPyEigenvectorSubspace)

I haven't checked if this works explicitly though by finding the linear combination that makes the two subspaces equivalent.

来源：https://stackoverflow.com/questions/25723478/difference-in-eigenvector-transformations-mathematica-vs-scipy

标签

python

scipy

wolfram-mathematica

linear-algebra