# Practical Uses for Eigenvalues

## Main Question or Discussion Point

I am trying to get some intuition for Eigenvalues/Eigenvectors. One real-life application appears to be a representation of resonance.

What are some practical uses for Eigenvalues?

What other things may Eigenvalues represent?

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HallsofIvy
Homework Helper
Generally speaking, eigenvalues and eigenvectors allow us to "reduce" a linear operation to separate, simpler, problems. For example, if a stress is applied to a "plastic" solid, the deformation can be dissected into "principle direction"s- those directions in which the deformation is greatest. Vectors in the principle directions are the eigenvectors and the percentage deformation in each principle direction is the corresponding eigenvalue.

If you have a bunch of data points, and you form the covariance matrix, then the the first eigenvector is normal to the least squares line (in 2D) or plane( in 3D) of that data. Thus you can use the eigenvector to find the least squares plane of some data, or the least squares line, or approximate the surface normal of a point cloud, or find the edges in an image (eg, with the structure tensor). The eigenvalue is the least squared error of the fit. In the Harris corner detector (for images), corners are detected by looking at the ratio between eigenvalues.

My favorite illustration of the usefulness of eigenvalues comes from probability. Suppose you represented the "state" of a ping-pong game by a vector, where the x-coordinate was the probability that I was serving and the y-coordinate was the probability that you were serving. It's possible to model the outcome of a serve by multiplying this vector by a matrix. (Not all real life situations can be modeled accurately this way, of course.)

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

$$v = \lambda_1^n e_1 + \lambda_2^n e_2 = e_1$$ 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.

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