On Fri, Jan 15, 2010 at 11:32 AM, Sebastian Walter <sebastian.wal...@gmail.com> wrote: > numpy.linalg.eig guarantees to return right eigenvectors. > evec is not necessarily an orthonormal matrix when there are > eigenvalues with multiplicity >1. > For symmetrical matrices you'll have mutually orthogonal eigenspaces > but each eigenspace might be spanned by > vectors that are not orthogonal to each other. > > Your omega has eigenvalue 1 with multiplicity 3.
Yes, I thought about the multiplicity. However, even for random symmetric matrices, I don't get the result I change the example matrix to omega0 = np.random.randn(20,8) omega = np.dot(omega0.T, omega0) print np.max(np.abs(omega == omega.T)) I have been playing with left and right eigenvectors, but I cannot figure out how I could compose my original matrix with them either. I checked with wikipedia, to make sure I remember my (basic) linear algebra http://en.wikipedia.org/wiki/Eigendecomposition_(matrix)#Symmetric_matrices The left and right eigenvectors are almost orthogonal ev, evecl, evecr = sp.linalg.eig(omega, left=1, right=1) >>> np.abs(np.dot(evecl.T, evecl) - np.eye(8))>1e-10 >>> np.abs(np.dot(evecr.T, evecr) - np.eye(8))>1e-10 shows three non-orthogonal pairs >>> ev array([ 6.27688862, 8.45055356, 15.03789945, 19.55477818, 20.33315408, 24.58589363, 28.71796764, 42.88603728]) I always thought eigenvectors are always orthogonal, at least in the case without multiple roots I had assumed that eig will treat symmetric matrices in the same way as eigh. Since I'm mostly or always working with symmetric matrices, I will stick to eigh which does what I expect. Still, I'm currently not able to reproduce any of the composition result on the wikipedia page with linalg.eig which is puzzling. Josef > > > > > On Fri, Jan 15, 2010 at 4:31 PM, <josef.p...@gmail.com> wrote: >> I had a problem because linal.eig doesn't rebuild the original matrix, >> linalg.eigh does, see script below >> >> Whats the trick with linalg.eig to get the original (or the inverse) >> back ? None of my variations on the formulas worked. >> >> Thanks, >> Josef >> >> >> import numpy as np >> import scipy as sp >> import scipy.linalg >> >> omega = np.array([[ 6., 2., 2., 0., 0., 3., 0., 0.], >> [ 2., 6., 2., 3., 0., 0., 3., 0.], >> [ 2., 2., 6., 0., 3., 0., 0., 3.], >> [ 0., 3., 0., 6., 2., 0., 3., 0.], >> [ 0., 0., 3., 2., 6., 0., 0., 3.], >> [ 3., 0., 0., 0., 0., 6., 2., 2.], >> [ 0., 3., 0., 3., 0., 2., 6., 2.], >> [ 0., 0., 3., 0., 3., 2., 2., 6.]]) >> >> for fun in [np.linalg.eig, np.linalg.eigh, sp.linalg.eig, sp.linalg.eigh]: >> print fun.__module__, fun >> ev, evec = fun(omega) >> omegainv = np.dot(evec, (1/ev * evec).T) >> omegainv2 = np.linalg.inv(omega) >> omegacomp = np.dot(evec, (ev * evec).T) >> print 'composition', >> print np.max(np.abs(omegacomp - omega)) >> print 'inverse', >> print np.max(np.abs(omegainv - omegainv2)) >> >> this prints: >> >> numpy.linalg.linalg <function eig at 0x017EDDF0> >> composition 0.405241032278 >> inverse 0.405241032278 >> >> numpy.linalg.linalg <function eigh at 0x017EDE30> >> composition 3.5527136788e-015 >> inverse 7.21644966006e-016 >> >> scipy.linalg.decomp <function eig at 0x01DB14F0> >> composition 0.238386662463 >> inverse 0.238386662463 >> >> scipy.linalg.decomp <function eigh at 0x01DB1530> >> composition 3.99680288865e-015 >> inverse 4.99600361081e-016 >> _______________________________________________ >> NumPy-Discussion mailing list >> NumPy-Discussion@scipy.org >> http://mail.scipy.org/mailman/listinfo/numpy-discussion >> > _______________________________________________ > NumPy-Discussion mailing list > NumPy-Discussion@scipy.org > http://mail.scipy.org/mailman/listinfo/numpy-discussion > _______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
