What Does a Strong Eigenvalue Signify in a System of Equations?

[LouArnold]

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?

 

[ThirstyDog]

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
$${y_{1} \choose y_{2}} = \alpha {1 \choose 2} e^{4t} + \beta {3 \choose 5} e^{5t}$$
So 4 is an eigenvalue associated with the eigenvector ${1 \choose 2}$ and 5 is the eigenvalue associated with the eigenvector ${3 \choose 5}$.

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

In any system the largest eigenvalue will lead to its associated eigenvector dominating as time tends to infinity.

 

[LouArnold]

In any system the largest eigenvalue will lead to its associated eigenvector dominating as time tends to infinity.

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?

 

[HallsofIvy]

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.