Eigenvalues and eigenvectors can be thought of in two complementary ways: a defining equation, and a matrix factorization.
A lot of intuition about eigenvalues and eigenvectors can come from considering the defining equation $Ax=\lambda x$. E.g., when $x$ is an eigenvector of $A$, operation by $A$ has the effect of scaling $x$.
Eigenvalues and eigenvectors can also be thought of (when $A$ is symmetric) as a matrix factorization that can be written $Q \Lambda Q^T$, where the columns of $Q$ are eigenvectors and $\Lambda$ is a diagonal matrix of eigenvalues.
The SVD is invariably given as a matrix factorization $U \Sigma V^*$. So, is there a defining equation for the SVD corresponding to the equation $Ax=\lambda x$ enjoyed by eigenvalues and eigenvectors?