# Eigenvalues / Eigenvectors relationship to Matrix Entries Values

1. Jun 10, 2014

### 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,

2. Jun 10, 2014

### AlephZero

3. Jun 10, 2014

### jorgejgleandro

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 gonna look into these theorems and return, in case they don't suit my needs.

Regards.

4. Jun 10, 2014

### 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.

5. Jun 10, 2014

### jorgejgleandro

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

6. Jun 10, 2014

### 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.