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Eigenvalues / Eigenvectors relationship to Matrix Entries Values

  1. Jun 10, 2014 #1
    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. jcsd
  3. Jun 10, 2014 #2

    AlephZero

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  4. Jun 10, 2014 #3
    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.
     
  5. Jun 10, 2014 #4

    hilbert2

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    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.
     
  6. Jun 10, 2014 #5
    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([itex]\lambda[/itex] * 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
     
  7. Jun 10, 2014 #6
    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.
     
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