Generalized Eigenvalue Problem for Three matrices

From: Sameer Agarwal (sandwichmaker_at_gmail.com)
Date: 10/30/04 

Date: 30 Oct 2004 10:02:03 -0700

Hello Everyone,

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.

The case when \lambda = 0 or \mu = 0 is an instance of the standard
generalized eigenvalue problem, and is discussed in the standard
texts, but I do not seem to be able to find anything on the above
problem. The problem is a slight simplification of the general
bilinear system

x^T P_i y = 0

where P_i are a collection of 3x3 matrices. I can make some
assumptions about the components of x not being zero, and hence work
with the ratios instead, which leads to the above mentioned eigenvalue
problem. About the only reference I could find on this is the
Technical Report by Cohen and Tomasi on "Systems of Bilinear
Equations", which says that the generalized eigenvalue problem for 3
or more matrices is unsolved, but it was written in 1994 and I am
wondering if there have been any developments since, or if there are
particular simplifications that occur as a result of the size of the
problem.

Any insights into the solution of the above problems or pointers to
the literature will be much appreciated.

Thanks,
Sameer