Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Meaning of strong eigenvalues

  1. Mar 27, 2009 #1
    In a system of equations with several eigenvalues, what does it mean (signify) when one is strong (high in value) and the others are weak (low in value)?

    Can a general statement be made without referencing an application? If so, is there a math book that explains the idea?
     
  2. jcsd
  3. Mar 27, 2009 #2
    If you are referring to a system of coupled ODE's then 'strong' eigenvalues correspond to the dominate eigenvectors. When I say dominate I mean ones that when time tends to infinity that the system follows a straight line given by the eigenvector.

    Suppose you have two coupled 1st order ODE's which yield the general solution
    [tex] {y_{1} \choose y_{2}} = \alpha {1 \choose 2} e^{4t} + \beta {3 \choose 5} e^{5t}[/tex]
    So 4 is an eigenvalue associated with the eigenvector [itex] {1 \choose 2} [/itex] and 5 is the eigenvalue associated with the eigenvector [itex] {3 \choose 5} [/itex].

    As time goes to infinity [itex] e^{5t} [/itex] becomes much larger than [itex] e^{4t} [/itex]. Thus we consider
    [tex] {y_{1} \choose y_{2}} \approx \beta {3 \choose 5} e^{5t}, [/tex]
    which leads to
    [tex] \frac{y_{1}}{y_{2}} \approx \frac{3}{5} \Rightarrow y_{2} \approx \frac{5}{3}y_{2}. [/tex]

    In any system the largest eigenvalue will lead to its associated eigenvector dominating as time tends to infinity.
     
  4. Mar 29, 2009 #3
    I understand the explanation in the way that it applied to ODE. The solution to the system is clear. But I was thinking of a least square problem. Whether SVD or the standard eigenvalue calculation is used, what is the significance of larger versus smaller eigenvalues?
     
  5. Mar 29, 2009 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    The only thing I can think of is that, generally, it is easier to numerically find a large (in absolute value) eigenvalue than a smaller. Numerical methods typically find the largest eigenvalue, then remove that eigenvalue and apply the same method to find the next largest eigenvalue.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Meaning of strong eigenvalues
  1. Strongly nilpotent (Replies: 1)

Loading...