I Normal Modes: Finding Eigenfrequencies

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To find the eigenfrequencies corresponding to the given eigenvectors of a system with three particles, one must typically solve the characteristic equation derived from the system's dynamical matrix. The eigenfrequencies are determined by the square roots of the eigenvalues obtained from this matrix. The discussion highlights a lack of clarity in the source material, which provided eigenfrequencies without explaining the derivation process. This has led to confusion about the method to calculate them. Understanding the relationship between eigenvectors and eigenfrequencies is crucial for analyzing the system's motion.
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If I have a system where the following is found to describe the motion of three particles:

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The normal modes of the system are given by the following eigenvectors: $$(1,0,-1), (1,1,1), (1,-2,1)$$
How can I find the corresponding eigenfrequencies? It should be simple... What am I missing?
 
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Is this for schoolwork?
 
berkeman said:
Is this for schoolwork?
This is something I found in a pdf online where it simply asserted what the eigenfrequencies were... and not how to find them.
 
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