Dear,

Please find attached something i wrote.

On top of what i wrote, i think the small eigenvalues are the ones becoming unstable. 

But please answer the question i've asked you before we go on

Hello Dear,

Thanks for your prompt response, please find the attached file where I explained the problem in detail. Hope you will find the answer to your question there.

Dear Muhammad,

Consider the following text book:

G.W. Stewart, Matrix Algorithms Volume II: Eigensystems, SIAM, Philadelphia, 2001. pp.130-131

There are some square matrices A and B of order 2, if B is singular, it is possible for any number λ to be an eigenvalue of the pencil   λB  - A.

The generalized eigenvalue problems are dependent on the pair (A,B).

     In your problem is an accident, because if we change the pair (A,E) to another pair, the phenomenon may be change.

Dear Muhammad, 

I have mistakenly read Identity instead of All-ones.

My sincere apologies.

Hello,

I think this is a low rank perturbation problem. In other words, the matrix pencil is perturbed by matrices of low rank and we have to find the perturbed eigenvalues and the eigenvectors. I found a paper that deals with the solution of the low rank perturbation problem in general. In my case, matrix A is perturbed by a rank 1 matrix J and E is not perturbed at all. I think we can relate this rank 1 perturbation problem to the problem discussed in this paper. Please find the attached paper. We can proceed further by discussing about the paper and try to find the solution of my problem in light of the paper.

Thanks 

Muhammad

It is a good idea, I am agree with your idea, Sir.