Physical meaning of multiple eigenvalues/vectors

tomwaits

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I'm tackling this problem outside of formal study, any approach is acceptable.

I found the solution of a dynamic system in the form \(\displaystyle x=\hat{x} \cdot e^{\sigma t + iky}\), where by differentiation I get a linear system to which the solutions are eigenvalues \(\displaystyle \sigma\) and eigenvectors \(\displaystyle \hat{x}\).

I have 4 initial equations, implying a 4x4 matrix resulting in 4 linearly independent eigenvalues/vectors. Suppose two of the eigenvalues are \(\displaystyle \sigma=0 + \pm 2i\) and the other two are \(\displaystyle \sigma = \pm 0.2+ 0i\).

Im confused as to what these solutions physically imply. Individually considered, I understand that the mode with eigenvalue \(\displaystyle \sigma=-0.2\) rapidly decays to zero, the mode with eigenvalue \(\displaystyle \sigma=+0.2\) implies exponential growth and the modes \(\displaystyle \sigma = \pm 2i\) imply rotation about a unit circle. However, I do not understand if the physical system 'chooses' one of these modes (by an initial condition, for example) or whether the physics is governed by superposition of all of the eigenvalues (i.e rotation + exponential growth)?
 
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I think I responded to this before- though it may have been on a different forum. No, the system does not "choose" one of the solutions. The general solution is a linear combination of all independent solutions with undetermined coefficients. Those coefficients, and the evolution of the system, is determined by specific initial conditions.
 
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