## Franklin's Notes

### Rayleigh quotient

Given a vector $x\in\mathbb R^m$ and a matrix $A\in\mathbb R^{m\times m}$, the Rayleigh quotient $r(x)$ is defined as the quantity Note that if $v$ is an eigenvector of $A$, then $r(v)=\lambda$ is the corresponding eigenvalue. If $r$ is viewed as a continuous function on the unit sphere in $x\in\mathbb R^m$, then the "stationary points" of $r$ are the normalized eigenvectors of $A$: for we have so that $\nabla r(v)= 0$ for eigenvectors $v$. This implies that for $v$ an eigenvector, and $x$ ranging over the unit sphere, and therefore $r(x)$ is quadratically accurate as an eigenvalue estimate as $x$ approaches the true eigenvector $v$. This fact is the basis for the power iteration algorithm.