Re: Generalized Eigenvalue Problem for Three matrices

From: Robert Israel (israel_at_math.ubc.ca)
Date: 10/31/04 

Date: 31 Oct 2004 07:50:14 GMT

In article <d93ed69a.0410300902.15980e0d@posting.google.com>,
Sameer Agarwal <sandwichmaker@gmail.com> wrote:

>In my work I have encountered a problem of the following form

>(A - \lambda B - \mu C) x = 0

>where x is an unknown 3x1 vector and A,B,C are nx3 matrices. The
>scalars \mu and \lambda are also unknown.

You want lambda and mu so that A - lambda B - mu C has rank
< 3. Now this is equivalent to saying that every 3 x 3 submatrix
has determinant 0. For any choice of the three rows,
the determinant of this submatrix is a polynomial in lambda and mu.
In the easy case, two of these polynomials will have a nonzero
resultant with respect to one of the variables, and then you have
a finite number of possibilities to check. In general, I think
you can use Groebner basis methods to find the solutions.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

Relevant Pages

  • Re: Generalized Eigenvalue Problem for Three matrices
    ... >the determinant of this submatrix is a polynomial in lambda and mu. ... two of these polynomials will have a nonzero ... The resultant of these wrt lambda is ...
    (sci.math)
  • expanding a determinant
    ... I am trying to expand the determinant of the following 2n x 2n matrix: ... nonsinglular n*n matrix, lambda is a ... scalar and I_n is the n*n identity matrix. ...
    (sci.math)
  • Berlekamp-Massey in Matlab
    ... Matlab, and I'm having a devil of a time. ... it'll calculate correct polynomials for a simple code (from a chapter ... % syndrome - Syndrome computed from received codeword ... convolve syndrome and previous lambda to get delta ...
    (comp.dsp)
  • Re: Matrix Minimax, sorta
    ... >fix a/b so that the determinant vanishes) this ... The rank is the size of the largest nonsingular ... square submatrix. ... solving a set of polynomials. ...
    (sci.math)