[Octave-bug-tracker] [bug #38577] Eig returns non-unitary transformation matrix

[Jordi Gutiérrez Hermoso]

Follow-up Comment #7, bug #38577 (project octave):

> Hermitian matrices can never be spectrally defective.

But they can be very close to defective. A defective matrix is somewhat
numerically invisible, since any perturbation destroys its defect. This is why
the Jordan form is somewhat useless, since it's numerically unstable.

> Eig returning a zero eigenvector for a hermitian matrix is a sign of a bug.


Not necessarily, a defective matrix can be close to being Hermitian, e.g. [1
eps; 0 1].

The eigenvalue routine that Octave uses isn't implemented in Octave. It's just
calling LAPACK's dgeev family of functions:

http://hg.savannah.gnu.org/hgweb/octave/file/a2f65b8f1955/liboctave/numeric/EIG.cc#l31

I'm not sure how your matrix is getting close to being defective, and it seems
to depend on the algorithm since the Schur factorisation doesn't exhibit this
problem. I have had reports that different Octave builds on Windows don't have
this problem.

I don't have a development machine right now to test on, but if you're able to
build Octave and step through it with a debugger, you should check if indeed
the LAPACK dseev symmetric eigenvalue algorithm is being called for your
matrix. If it isn't, then there's a possible bug in Octave. If it is, then the
problem is in LAPACK.

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