Rayleigh quotient

The single best number-guess for an eigenvalue you can squeeze out of a vector that's only approximately an eigenvector.

For a vector $\mathbf v\ne\mathbf 0$ , the Rayleigh quotient is $R(\mathbf v)=\dfrac{\mathbf v^{\mathsf T}A\mathbf v}{\mathbf v^{\mathsf T}\mathbf v}$ . It equals $\lambda$ exactly whenever $\mathbf v$ is an exact eigenvector for $\lambda$ , and it is the least-squares-optimal scalar estimate of an eigenvalue for any other $\mathbf v$ -- extracting an eigenvalue estimate from a power-iteration vector at every step, not only at convergence.

Examples

Even after just one power-iteration step, the Rayleigh quotient of the current iterate is usually a far better eigenvalue estimate than the raw iterate itself would suggest.