Eigen vectors in python giving seemingly random element-wise signs

Question

I'm running the following code:

import numpy as np
import matplotlib
matplotlib.use("TkAgg")
import matplotlib.pyplot as plt

N = 100
t = 1

a1 = np.full((N-1,), -t)
a2 = np.full((N,), 2*t)
Hamiltonian = np.diag(a1, -1) +  np.diag(a2) + np.diag(a1, 1)

eval, evec = np.linalg.eig(Hamiltonian)
idx = eval.argsort()[::-1]
eval, evec = eval[idx], evec[:,idx]

wave2 = evec[2] / np.sum(abs(evec[2]))
prob2 = evec[2]**2 / np.sum(evec[2]**2)

_ = plt.plot(wave2)
_ = plt.plot(prob2)
plt.show()

And the plot that comes out is this:

But I'd expect the blue line to be a sinoid as well. This has got me confused and I can't find what's causing the sudden sign changes. Plotting the function absolutely shows that the values associated with each x are fine, but the signs are screwed up.

Any ideas on what might cause this or how to solve it?

Answer 1

Here's a modified version of your script that does what you expected. The changes are:

• Corrected the indexing for the eigenvectors; they are the columns of evec .
• Use np.linalg.eigh instead of np.linalg.eig . This isn't strictly necessary, but you might as well use the more efficient code.
• Don't reverse the order of the sorted eigenvalues. I keep the eigenvalues sorted from lowest to highest. Because eigh returns the eigenvalues in ascending order, I just commented out the code that sorts the eigenvalues.

(Only the first change is a required correction.)

---

import numpy as np
import matplotlib
import matplotlib.pyplot as plt

N = 100
t = 1

a1 = np.full((N-1,), -t)
a2 = np.full((N,), 2*t)
Hamiltonian = np.diag(a1, -1) +  np.diag(a2) + np.diag(a1, 1)

eval, evec = np.linalg.eigh(Hamiltonian)
#idx = eval.argsort()[::-1]
#eval, evec = eval[idx], evec[:,idx]

k = 2
wave2 = evec[:, k] / np.sum(abs(evec[:, k]))
prob2 = evec[:, k]**2 / np.sum(evec[:, k]**2)

_ = plt.plot(wave2)
_ = plt.plot(prob2)
plt.show()

The plot:

Answer 2

I may be wrong, but aren't they all valid eigen vectors/values? The sign shouldn't matter, as the definition of an eigen vector is:

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that only changes by an overall scale when that linear transformation is applied to it.

Just because the scale is negative doesn't mean it isn't valid.

See this post about Matlab's eig that has a similar problem

One way to fix this is to simply pick a sign for the start, and multiply everthing by -1 that doesn't fit that sign (or take abs of every element and multiply by your expected sign). For your results this should work (nothing crosses 0).

Neither matlab nor numpy care about what you are trying to solve, its simple mathematics that dictates that both signed eigenvector/value combinations are valid, your values are sinusoidal, its just that there exists two sets of eigenvector/values that work (negative and positive)