An eigenvector is a vector in the preimage of the transformation whose direction is not changed when the transformation is applied. It seems like the concept of eigenvectors and eigenvalues would extend beyond matrices (linear transformations) to vector fields (nonlinear transformations). For example, for the vector field $\langle y^2,x^2\rangle$, any vector whose components are identical is an eigenvector of this vector field (if the term applies here). I only figured this by observation, but is there a general way to find these "eigenvectors" for a vector field? Also, it seems the "eigenvalues" would be a function rather than a finite set (e.g. the eigenvalue for $\langle1,1\rangle$ is 1, but the eigenvalue for $\langle3,3\rangle$ is 3, and so on).
What is the interpretation of this? What uses does/might it have in mathematics or engineering?