[SciPy-Dev] Generalized eigenproblem with rank deficient matrices

Charles R Harris charlesr.harris@gmail....
Sun Sep 4 11:55:43 CDT 2011


On Sun, Sep 4, 2011 at 10:09 AM, Nils Wagner
<nwagner@iam.uni-stuttgart.de>wrote:

> On Sun, 4 Sep 2011 09:29:19 -0600
>  Charles R Harris <charlesr.harris@gmail.com> wrote:
> > On Sun, Sep 4, 2011 at 7:53 AM, Nils Wagner
> ><nwagner@iam.uni-stuttgart.de>wrote:
> >
> >> Hi all,
> >>
> >> how can I solve the eigenproblem
> >>
> >> A x = \lambda B x
> >>
> >> where both matrices are rank deficient ?
> >>
> >
> > I'd do eigh and transform the problem to something like:
> >
> > U * A  * U^t * x= \lambda D * x
> >
> > where D is diagonal. Note that the solutions may not be
> >unique and \lambda
> > can be arbitrary, as you can see by studying
> >
> > A = B = array([[1, 0], [0, 0]])
> >
> > Where there are solutions for arbitrary \lambda.
> >Likewise, there may be no
> > solutions under the requirement that x is non-zero:
> >
> > A = array([[1, 1], [1, 0]]),
> > B = array([[1, 0], [0, 0]])
> >
> > The usual case where B is positive definite corresponds
> >to finding extrema
> > on a compact surface x^t * B *x = 1, but the surface is
> >no longer compact
> > when B isn't positive definite. Note that these cases
> >are all sensitive to
> > roundoff error.
> >
> > Chuck
>
> Hi Chuck,
>
> I am only interested in the real and complex
> eigensolutions.
> The complex eigenvalues appear in pairs  a_i \pm \sqrt{-1}
> b_i sind A and B are real.
> How can I reject infinite eigenvalues ?
> Both matrices, A and B, are indefinite.
>
>
It depends on the particular problem. In general, the solution to the
generalized eigenvalue problem starts by making a variable substitution that
reduces B to the identity matrix, usually by using a Cholesky factorization,
i.e., B = U^t U, y = U x. This can still be done in the (numerically)
indefinite case but Cholesky won't be reliable and that is why I suggested
eigh. Note that the problem we are trying to solve is
finding the extrema of x^t A x subject to the constraint x^t B x = 1,
\lambda is then a Lagrange  multiplier. I assume A and B are both symmetric?
Anyway, in terms of y, the problem then reduces to finding extrema of y^t
U^t^{-1} A U^{-1} y subject to y^t D y = 1, where D is diagonal and has ones
along part of the diagonal, zeros for the remainder. In the usual case, D is
the identity. The trick is then to divide y into two parts, one for where D
is one (u), another for the rest (v), so that y = [u v]. If you are lucky,
the v can be solved in terms of u using the transformed A, and things will
reduce to an eigenvalue problem for u. If the original A was symmetric, so
will be the reduced problem. If you can't solve v as a function of u, then
you can reduce things further, but it is possible that at some point there
is no solution.

I don't have practical experience with this sort of problem with indefinite
B, so I can't tell you much more than that. I assume you've googled for
relevant documents.

Chuck
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