# Meaning of strong eigenvalues

## Main Question or Discussion Point

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?

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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
$${y_{1} \choose y_{2}} \approx \beta {3 \choose 5} e^{5t},$$
$$\frac{y_{1}}{y_{2}} \approx \frac{3}{5} \Rightarrow y_{2} \approx \frac{5}{3}y_{2}.$$

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

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