Is scipy.linalg.eig giving the correct left eigenvectors?

About vl, the eig docstring says:

a.H vl[:,i] = w[i].conj() b.H vl[:,i]

Or, taking the conjugate transpose (i.e. Hermitian transpose) of both sides (which is what .H means), and assuming b is the identity,

vl[:,i].H a = w[i] vl[:,i].H

So the rows of the conjugate transpose of vl are the actual left eigenvectors of a.

Numpy arrays don't actually have the .H attribute, so you must use .conj().T.

Here's a script to verify the calculation:

import numpy as npfrom scipy.linalg import eig# This only affects the printed output.np.set_printoptions(precision=4)a = np.array([[6, 2],              [-1, 4]])w, vl, vr = eig(a, left=True)print "eigenvalues:", wprint# check the left eigenvectors one-by-one:for k in range(a.shape[0]):    val = w[k]    # Use a slice to maintain shape; vec is a 2x1 array.    # That allows a meaningful transpose using .T.    vec = vl[:, k:k+1]    # rowvec is 1x2; it is the conjugate transpose of vec.    # This should be the left eigenvector.    rowvec = vec.conj().T    # Verify that rowvec is a left eigenvector    lhs = rowvec.dot(a)    rhs = val * rowvec    print "Compare", lhs, "to", rhs    print rowvec, "is",    if not np.allclose(lhs, rhs):        print "*NOT*",    print "a left eigenvector for eigenvalue", valprintprint "Matrix version:"print "This"print vl.conj().T.dot(a)print "should equal this"print np.diag(w).dot(vl.conj().T)

Output:

eigenvalues: [ 5.+1.j  5.-1.j]Compare [[ 1.6330+2.4495j  4.0825+0.8165j]] to [[ 1.6330+2.4495j  4.0825+0.8165j]][[ 0.4082+0.4082j  0.8165-0.j    ]] is a left eigenvector for eigenvalue (5+1j)Compare [[ 1.6330-2.4495j  4.0825-0.8165j]] to [[ 1.6330-2.4495j  4.0825-0.8165j]][[ 0.4082-0.4082j  0.8165+0.j    ]] is a left eigenvector for eigenvalue (5-1j)Matrix version:This[[ 1.6330+2.4495j  4.0825+0.8165j] [ 1.6330-2.4495j  4.0825-0.8165j]]should equal this[[ 1.6330+2.4495j  4.0825+0.8165j] [ 1.6330-2.4495j  4.0825-0.8165j]]

Now, the eig docstring also says in the description of the return values:

vl : double or complex ndarray    The normalized left eigenvector corresponding to the eigenvalue    ``w[i]`` is the column v[:,i]. Only returned if ``left=True``.    Of shape ``(M, M)``.

and that is potentially misleading, since the conventional definition of a left eigenvector (e.g. http://mathworld.wolfram.com/LeftEigenvector.html or http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#Left_and_right_eigenvectors) is a row vector, so it is the conjugate transpose of the column of vl that is actually the left eigenvector.