Re: Understanding eig()

[Paul Kienzle]

On Aug 1, 2004, at 5:47 PM, Rich Shepard wrote:

> On Sun, 1 Aug 2004, Henry F. Mollet wrote:
>
> > In the real world we need real solutions but there are exceptions.  I 
> > assume
> > that GNU Octave lists the eigenvalues in the order of the absolute 
> > value
> > octave:3> la=diag(la)
> > la =
> >
> >    11.72374 +  0.00000i
> >    -1.05925 +  5.81855i
> >    -1.05925 -  5.81855i
> >    -0.00571 +  2.32593i
> >    -0.00571 -  2.32593i
> >    -2.03018 +  0.00000i
> >     0.21818 +  0.82953i
> >     0.21818 -  0.82953i
>
> Henry,
>
>   Then my assumption that what octave produced as a result of the eig()
> function is a list of the eigenvalues, in rank order, is correct. 
> That's
> good to know. Many thanks.

Underlying, Octave uses the routine dgeev, zgeev, zheev or dsyev
to compute the eigenvalues (depending on the type of input matrix).

I don't see any mention of rank ordering of eigenvalues on their
man pages (assuming you mean ranking by magnitude).  Indeed,
the following is a counter-example:

octave> abs(eig([4,5,0;-2,4,0;0,0,10]))
ans =

   5.0990
   5.0990
  10.0000

octave> abs(eig([4,5,0;-2,4,0;0,0,1]))
ans =

  5.0990
  5.0990
  1.0000

The only guarantee I read is that complex conjugates would
come in pairs. with the +i component first.

If you want to order the eigenvalues by decreasing magnitude, use:

        lambda = eig(A);
        [junk,idx] = sort(abs(lambda));
        lambda = lambda(flipud(idx));

If you want to order the eigenvalues and eigenvectors, use:

        [v,l] = eig(A);
        lambda = diag(l);
        [junk,idx] = sort(abs(lambda));
        v = v(:,flipud(idx));
        lambda = lambda(flipud(idx));

Since Octave uses a stable sort, this will preserve the
complex conjugate pair guarantee, by with the -i component
first since the list is reversed.

Hope that helps,

- Paul



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