Rayleigh quotient

Definition

Let $A$ be a Hermitian $n \times n$ complex matrix, and let $x \in \C^n$ with $x \ne 0$. The Rayleigh quotient of $A$ and $x$ is the real number $$R(A, x) = \frac{x^* A x}{x^* x}.$$

Remarks

For each $x \ne 0$, $\lambda_{\min} \le R(A, x) \le \lambda_{\max}$, where $\lambda_{\min}$ and $\lambda_{\max}$ are the least and the greatest eigenvalues of $A$. If $x$ is an eigenvector for $\lambda$, then $R(A, x) = \lambda$. So $\lambda_{\max} = \max_{x \ne 0} R(A, x)$ and $\lambda_{\min} = \min_{x \ne 0} R(A, x)$.

Explanations

L1 human

The Rayleigh quotient measures how much a symmetric matrix stretches a given direction, on average. Its largest value over all directions is the largest eigenvalue. This gives a way to estimate eigenvalues without solving the characteristic equation.