My be this + studies of Voevodin

Numerical Analysis and Its Applications

Lecture Notes in Computer Science Volume 1196, 1997, pp 192-199

A method for solving the spectral problem for complex matrices

Ivan Ganchev Ivanov

Thanks Dr Joseph, this is helpful. I was also refered to the Cauchy interface theorem is relevant (pointed to me by one of my colleagues), However, I was trying to find a closed form relationship that related eigen decompositions of the N+1xN+1 matrix to the NxN matrix. Not sure if it exist ast least when A and B are Gaussian kernels . I am checking the reference but

http://matrix.skku.ac.kr/Series-E/Monthly-E.pdf

For the eigenvalues there exist the beautiful interlacing results: if a_j, j=1,…,n+1, are the eigenvalues of A ordered in nondecreasing way and b_j, j=1,…,n, are the eigenvalues of B ordered in nondecreasing way, then b_j belongs to the interval [a_j,a_(j+1)] , , j=1,…,n.

These bounds could be quite precise when A is the approximation/discretization of a differential/integral operator, because the eigenvalues often behave as the sampling of smooth functions on compact domain.

The interlacing results can be proved using the min-max or max-min characterization (see for instance the book by Rajendra Bhatia, Matrix Analysis, Springer). These results apply not only to Hermitian positive definite matrices, but to Hermitian matrices.

Unfortunately, the behavior of the eigenvectors may be much more ‘irregular’

What can be said about cond(A) and cond(B) ?

In the positive definite case, given the interlacing results, cond(A) is larger than cond(B). On the other hand, you can construct examples where cond(A)/cond(B) is as large as you want.

Dear Dr Stefano !

1. Thank you for your answer.

2. Can you recommend some literature about impact of matrix transformations on

the condition number ?

Joseph

Dear Joseph I answered to you by email

I am really so grateful for this interesting information you shared, Dr Stefano and Dr Joseph.

Schur Decomposition: Given A∈Mn with eigenvalues ¸ λ₁,λ₂,…,λn in any prescribed order, there is a unitary matrix U∈Mn such that U∗AU is an upper triangular, with diagonal entries t_{ii} = λi¸ i=1,2,…,n.

     Let B∈Mn+1 with eigenvalues ¸ λ₁,λ₂,…,λn,λn+1, and A be a principal submatrix of order n of the positive definite matrix B. Then all eigenvalues of A is also eigenvlues of B.