Eigenanalysis of complex hermitian matrix: different phase angles for EIG and EIGH

Question

I understand that eigenvectors are only defined up to a multiplicative constant. As far as I see all numpy algorithms (e.g. linalg.eig, linalg.eigh, linalg.svd) yield identical eigenvectors for real matrices, so apparently they use the same normalization. In the case of a complex matrix, however, the algorithms yield different results.

That is, the eigenvectors are the same up to a (complex) constant z. After some experimenting with eig and eigh I realised that eigh always sets the phase angle (defined as arctan(complex part/real part)) to 0 for the first component of each eigenvector whereas eig seems to start with some (arbitrary ?) non-zero phase angle.

Q: Is there a way to normalize the eigenvectors from eigh in the way eig is doing it (that is not to force phase angle = 0)?

Example

I have a complex hermitian matrix G for which I want to calculate the eigenvectors using the two following algorithms:

• numpy.linalg.eig for a real/complex square matrix
• numpy.linalg.eighfor a real symmetric/complex hermitian matrix (special case of 1.)

Check that G is hermitian

# check if a matrix is hermitian
def isHermitian(a, rtol=1e-05, atol=1e-08):
    return np.allclose(a, a.conjugate().T, rtol=rtol, atol=atol)

print('G is hermitian:', isHermitian(G))

Out:

G is hermitian: True

Perform eigenanalysis

# eigenvectors from EIG()
l1,u1 = np.linalg.eig(G)
idx = np.argsort(l1)[::-1]
l1,u1 = l1[idx].real,u1[:,idx]

# eigenvectors from EIGH()
l2,u2 = np.linalg.eigh(G)
idx = np.argsort(l2)[::-1]
l2,u2 = l2[idx],u2[:,idx]

Check eigenvalues

print('Eigenvalues')
print('eig\t:',l1[:3])
print('eigh\t:',l2[:3])

Out:

Eigenvalues
eig     : [2.55621629e+03 3.48520440e+00 3.16452447e-02]
eigh    : [2.55621629e+03 3.48520440e+00 3.16452447e-02]

Both methods yield the same eigenvectors.

Check eigenvectors

Now look at the eigenvectors (e.g. 3. eigenvector) , which differ by a constant factor z.

multFactors = u1[:,2]/u2[:,2]
if np.count_nonzero(multFactors[0] == multFactors):
    print("All multiplication factors are same:", multFactors[0])
else:
    print("Multiplication factors are different.")

Out:

All multiplication factors are same: (-0.8916113627685007+0.45280147727156245j)

Check phase angle

Now check the phase angle for the first component of the 3. eigenvector:

print('Phase angel (in PI) for first point:')
print('Eig\t:',np.arctan2(u1[0,2].imag,u1[0,2].real)/np.pi)
print('Eigh\t:',np.arctan2(u2[0,2].imag,u2[0,2].real)/np.pi)

Out:

Phase angel (in PI) for first point:
Eig     : 0.8504246311627189
Eigh    : 0.0

Code to reproduce figure

num = 2
fig = plt.figure()
gs = gridspec.GridSpec(2, 3) 
ax0 = plt.subplot(gs[0,0])
ax1 = plt.subplot(gs[1,0])
ax2 = plt.subplot(gs[0,1:])
ax3 = plt.subplot(gs[1,1:])
ax2r= ax2.twinx()
ax3r= ax3.twinx()
ax0.imshow(G.real,vmin=-30,vmax=30,cmap='RdGy')
ax1.imshow(G.imag,vmin=-30,vmax=30,cmap='RdGy')
ax2.plot(u1[:,num].real,label='eig')
ax2.plot((u2[:,num]).real,label='eigh')
ax3.plot(u1[:,num].imag,label='eig')
ax3.plot((u2[:,num]).imag,label='eigh')
for a in [ax0,ax1,ax2,ax3]:
    a.set_xticks([])
    a.set_yticks([])
ax0.set_title('Re(G)')
ax1.set_title('Im(G)')
ax2.set_title('Re('+str(num+1)+'. Eigenvector)')
ax3.set_title('Im('+str(num+1)+'. Eigenvector)')
ax2.legend(loc=0)
ax3.legend(loc=0)
fig.subplots_adjust(wspace=0, hspace=.2,top=.9) 
fig.suptitle('Eigenanalysis of Hermitian Matrix G',size=16)
plt.show()

Answer 1

In my experience (and there are many questions here to back this up), you NEVER want to use eig when eigh is an option - eig is very slow and very unstable. The relevance of this is that I believe your question is backward - you want to normalize the eigenvectors of eig to be like those of eigh, and this you know how to do.