Rayleigh Quotient of a Self-Adjoint Operator

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space}, let $T$ be a \reftext{def:linear-operator-2026a}{linear operator} on $V$ that is \reftext{def:self-adjoint-operator-2026b}{self-adjoint}, and let $S$ be the set of \reftext{def:unit-vector-2026a}{unit vectors} of $V$.

The \textbf{Rayleigh quotient} of $T$ is the map $R_{T}:S\to\mathbb{R}$ sending each $x\in S$ to

which is a \reftext{def:real-numbers-c54-2026c}{real number} by claim 1 of \ref{lem:self-adjoint-elementary-properties-2026a}.