[dougabug]

The point of an eigenvector is that its direction is stable when you hit it with the given transformation. That is, it's an invariant direction of your transformation. This is analogous to the derivative of an exponential function being a simple multiple of that function. It makes it easier to reason about the action of the matrix, and simpler to represent.

Imagine I have a matrix transformation that takes a given input vector and converts it into a superposition of a thousand other vectors in random directions; that's far harder to reason about than if if just kicked it farther or brought it closer in the same direction.