Re: maximization of (generalized) eigenvector problem
- From: Jack Tomsky <jtomsky@xxxxxxxxxxxxx>
- Date: Mon, 15 Aug 2005 19:16:11 EDT
> Hi,
>
> I want to solve this problem, which looks familiar to
> me but with a
> little tweak:
>
> A and B are real symmetric semipositive definite
> matrices, both are p
> by p.
>
> I want to find the vectors vi, i=1 to p, so that the
> following is
> maximized:
>
> sum vi'*A*vi
> -------------
> sum vi'*B*vi
>
> , where the summation goes from i = 1 to p.
>
> What bothers me is the summation. Without the
> summation, it is a
> generalized eigenvalue problem. I wonder if the
> maximum solution is
> simply setting v1 to satisfy A*v1 = B*v1, and setting
> all the other vi
> to zero vectors?
>
There are two cases.
Case 1. If B is singular, find a vector v1 such that B*v1 = 0 and A*v1 is not 0. If no such vector exists, go to case 2.
Set vi = 0 for i>1.
Then the ratio of the sum of quadratic forms is infinite.
Case 2. If B is nonsingular, find an eigenvector v1 corresponding to the maximum eigenvalue of A*B^(-1). Call this maximum eigenvalue chmax.
Set the remaining vi = 0. Then the ratio of the sum of quadratic forms is chmax.
Jack
.
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