Eigenvalues / Eigenvectors relationship to Matrix Entries Values

[jorgejgleandro]

Hi, folks

I have had a hard time to find out whether or not there is a theorem in Linear Algebra or Spectral Theory that makes any strong statement about the relationship between the entries of a Matrix and its Eigenvalues and Eigenvectors.

Indeed, I would like to know how is the dependence between a matrix entries and its eigenvalues / eigenvectors. It could be something describing:
- what are the properties of the eigenvalues and eigenvectors of a matrix whose entries are less than 1 and greater than 0.
- what are the properties of the eigenvalues and eigenvectors of a matrix whose entries modulus are less than 1
- what are the properties of the eigenvalues and eigenvectors of a matrix whose entries goes to infinity

I'm studying spectral decomposition of matrices and would like to predict what will happen with the eigenvectors, given a diferent set of values for the Matrix entries.

I would appreciate any valuable reference with hints on that.

Regards,

 

[AlephZero]

This might be less than you hoped for, but it's a useful result: http://en.wikipedia.org/wiki/Gershgorin_circle_theorem

 

[jorgejgleandro]

This might be less than you hoped for, but it's a useful result: http://en.wikipedia.org/wiki/Gershgorin_circle_theorem

Thanks, AlephZero.
I've just had a quick glimpse over it and suspect it looks like being far more than I expect, indeed. However, among the links for similar theorems at the bottom of that page, there is the Perron–Frobenius theorem, which appears to be a good clue towards the right track.

I'm going to look into these theorems and return, in case they don't suit my needs.

Regards.

 

[hilbert2]

One interesting property of matrix eigenproblems is that a small change in the matrix entries can sometimes cause a large change in the eigenvectors and eigenvalues. Then we say that the eigensystem is unstable. What I mean is that the eigenvalues of, say, matrices
$\left[\begin{smallmatrix}1&2\\3&4\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}1&2.01\\3&4\end{smallmatrix}\right]$
are not necessarily close to each other. It can be shown that eigensystems with hermitian matrices are stable, though.

 

[jorgejgleandro]

One interesting property of matrix eigenproblems is that a small change in the matrix entries can sometimes cause a large change in the eigenvectors and eigenvalues. Then we say that the eigensystem is unstable. What I mean is that the eigenvalues of, say, matrices
$\left[\begin{smallmatrix}1&2\\3&4\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}1&2.01\\3&4\end{smallmatrix}\right]$
are not necessarily close to each other. It can be shown that eigensystems with hermitian matrices are stable, though.

Yeah, I'm after something like you've described, Hilbert2.

Let me describe the big picture of the question for which I've been looking for an answer.

Given a non-normalized Laplacian matrix L of a Graph (this matrix is a symmetric real-valued matrix - a special case of Hermitian matrices), which are known to have real eigenvalues and an orthonormal set of eigenvectors, I want to carry out a wavelet analysis on it through the SGWT of [Hammond et al, 2009]. After its spectral decomposition, it's possible to use the Functional Calculus to evaluate a kernel g(\lambda * t) in the spectral domain on every L eigenvalue.

I wish I could analytically demonstrate how the Spectral Graph Wavelet Transform values depend on the L matrix entries. However, I think it's necessary to show how the L eigenvalues depend on the L entries beforehand.

Regards

 

[Haborix]

The first idea that came to mind while reading your post was the Schur-Horn theorem. I'm not sure if it helps, but perhaps related literature will help guide you to what you need.