Ononot — The Rayleigh quotient of a vector \mathbf{x} for a symmetric

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Eigenvalues & Eigenvectorseasy

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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 $A$ ?