Core curriculum - Mathematics: Linear algebra VI

 

A13. Eigenvectors and Eigenvalues

Let's end this section on linear algebra with a brief exploration of eigenvectors and their eigenvalues. An eigenvector is simply one which is unchanged by a linear transformation except to be scaled by some constant. The constant factor (scalar) by which the eigenvector is scaled is called its eigenvalue. If the eigenvalue is negative then the direction of the vector is reversed as well as scaled.

 Curious about the terminology? Eigen means "proper" or "characteristic" in German. So if you're struggling to understand or remember what eigenvectors are all about, perhaps it helps to rename them "characteristic vectors" instead.

Here's a nice introduction to the concepts. Pay close attention to the symmetry arguments. It turns out eigenvectors represent things like axes of rotational symmetry and the like:

 

 

And with some of the insights under your belt, here's a tutorial on the mechanics of finding eigenvalues and eigenvectors:

 

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