Re: help on changes of eigenvalues

On Mon, 15 Oct 2007 05:53:36 -0700, katherina78 wrote:


Hi,

I want to rephrase my question in a previous post:

How can one find changes in eigenvalues of a negative definite nxn
matrix A, when the following occurs:

M(d):=AV(D^{-1}-I-V'AV)^{-1}V'A'

D=diag(d), d=(d1,...,dr) and 0<di<1, V is nxr and V'AV is negative
definite rxr matrix, I is rxr identity matrix. (Then (D^{-1}-I-
V'AV)^{-1} is a rxr positive definite matrix and in my examples M(d)
becomes a nxn positive semi-definite matrix)

What is the relation between eigenvalues and eigenvectors of A and
M(d)?

Indeed we have the result, A+M(d) is negative definite for di=0 and
negative semi-definite for di=1. A+M(d) negative semi-definite for
0<di<1?

Any idea about this problem?
Let B = -A and E = D^(-1)-I
then
M = BV(E+V'BV')^(-1)V'B
A+M(d) = -(B-BV(E+V'BV')^(-1)V'B)
= -(B^-1 + V'E^(-1)V)^-1 (if E is invertible)
Now if A<0 then B>0, and if 0<D<I then E = D^(-1)-I > 0
so B^-1 + V'E^(-1)V) > 0 and so A+M(d)<0
(Here B>0 means B is positive definite etc).



.

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