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What are some practical uses for Eigenvalues?

What other things may Eigenvalues represent?

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What are some practical uses for Eigenvalues?

What other things may Eigenvalues represent?

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HallsofIvy

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It turns out that one of the eigenvalues of the matrix will be exactly one, and the other will be less than one. (The proof that this must be so is not obvious, but it stems from the fact that probabilities always sum to one.) Think about what that means. It means that if we keep playing, we keep multiplying the state of the game by the matrix over and over again. The eigenvector corresponding to the smaller eigenvalue keeps getting multiplied by a smaller and smaller value, shrinking to insignificance. The other eigenvector keeps getting multiplied by one, unchanging. You can use this knowledge of the eigenvalues to predict what the long term behavior in the game will be... how frequently each of us will be serving.

If [itex]e_1[/itex] and [itex]e_2[/itex] are the eigenvectors and [itex]\lambda_1[/itex] and [itex]\lambda_2[/itex] represent the eigenvalues, then consider what happens when we multiply any linear combination of them by the matrix over and over again.

[tex]v = \lambda_1^n e_1 + \lambda_2^n e_2 = e_1[/tex] as n explodes

The neat thing is that the initial condition, whether you or I started the first serve, will fade to insignificance. All because one eigenvalue is one and the other is smaller than one.

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Good point about resonance.One real-life application appears to be a representation of resonance.

Eigenvalues can represent the fundamental modes of vibration of, say, a beam. So they might indicate when a bridge might experience destructive vibrations (collapse) due to wind, etc.

In the field of aerospace, a similar analysis might be done on the airfoil of an airplane for aeroelastic purposes (i.e. - to determine when flutter might occur).

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