The Rayleigh quotient of a vector $\mathbf{x}$ for a symmetric matrix $A$ is defined as $R(\mathbf{x}) = \frac{\mathbf{x}^T A \mathbf{x}}{\mathbf{x}^T \mathbf{x}}$. For which vectors does $R(\mathbf{x})$ equal an eigenvalue of $A$?