What do eigenvalues and eigenvectors represent intuitively?

Answer by Karolis Juodele:

Eigenvectors and eigenvalues are not something you use in a matrix. It's something a matrix has.

Eigenvector is a vector which, multiplied by the matrix does not change direction. It may change length. Eigenvalue of an eigenvector is the magnitude by which the vector length changed.

One example, the rotation matrix rotates all vectors except those that match its rotation axis. These vectors are its eigenvectors and their eigenvalues are 1. Another example could be a scaling matrix which doubles the length of any vector. All vectors are its eigenvectors and their eigenvalues are equal to 2.

What do eigenvalues and eigenvectors represent intuitively?

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