Clojure Linear Algebra Refresher (2) - Eigenvalues and Eigenvectors

If there are scalar \(\lambda\) and a non-zero vector \(\mathbf{x}\) such that \(A\mathbf{x} \lambda\mathbf{x}\), we call such scalar eigenvalue, and such vector eigenvector. There can be more than one eigenvalue for a given matrix, and there is an infinite number of eigenvectors corresponding to one eigenvalue. All eigenvectors that correspond to one eigenvalue lie on the same line, but have different magnitudes. Seems simple, and it is, but so what? It looks like a trivial thing; how come these eigenvectors and eigenvalues are so ubiquitous in linear algebra?