> Does anyone know of an example of a simple physical system where eigenvalues have a physical interpretation?
Yep, vibration modes. Vibration frequencies represent their eigenvalues while the shape that the structural system exhibits when subjected to said vibration corresponds to it's eigenvector.
If a structural system is modelled as a linear elastic system it's possible to apply an eigendecomposition of that system and represent it in terms of linear combinations of it's vibration modes/eigenvector, and consequently we can get very accurate representations by using only a hand-full of these eigenvectors.
You know swing sets? We would start to swing back and forth just by moving our legs in a particular frwquencey, and without much effort we could move more and more? It turns out the frequency we moved our legs was the system's vibration frequency/eigenvalue for the vibration modes/eigenvector representing the we swinging back and forth.
> Does this relate to the normal modes or eigenmodes of a system?
Yes. The eigenvalues and eigenvectors of an undamped harmonic oscillator are respectively the vibration frequency and vibration mode.
One major class of structural analysis techniques is modal analysis, which determines the vibration modes and corresponding frequencies of specific structural systems subjected to particular boundary conditions.
Yep, vibration modes. Vibration frequencies represent their eigenvalues while the shape that the structural system exhibits when subjected to said vibration corresponds to it's eigenvector.
If a structural system is modelled as a linear elastic system it's possible to apply an eigendecomposition of that system and represent it in terms of linear combinations of it's vibration modes/eigenvector, and consequently we can get very accurate representations by using only a hand-full of these eigenvectors.
You know swing sets? We would start to swing back and forth just by moving our legs in a particular frwquencey, and without much effort we could move more and more? It turns out the frequency we moved our legs was the system's vibration frequency/eigenvalue for the vibration modes/eigenvector representing the we swinging back and forth.