---
tag: J858GV245K
kind: definition
name: Rayleigh quotient for symmetric kernels
msc: [45C05, 47A75]
defines: HRCQ3JHYY0
uses: [WWMWX7AD4V, 2ZB3NFZ94T, CB5G212Q03, 06WHP8SE98]
sources:
  - url: https://en.wikipedia.org/wiki/Rayleigh_quotient
---

## Definition

Let $a < b$ and let $K \in L^2([a, b] \times [a, b])$ be a [[square-integrable kernel|WWMWX7AD4V]] that is [[Hermitian|2ZB3NFZ94T]]: $K(y, x) = \overline{K(x, y)}$ for almost all $x, y$.
Let $T_K$ be its [[Hilbert–Schmidt integral operator|CB5G212Q03]] on $L^2[a, b]$, and let $\ip{u}{v} = \int_a^b u(x) \overline{v(x)} \, dx$.
The *Rayleigh quotient* of $K$ at a function $\varphi \in L^2[a, b]$ that is not $0$ almost everywhere is
$$R_K(\varphi) = \frac{\ip{T_K \varphi}{\varphi}}{\ip{\varphi}{\varphi}} = \frac{\int_a^b \int_a^b K(x, y) \, \varphi(y) \, \overline{\varphi(x)} \, dx \, dy}{\int_a^b \abs{\varphi(x)}^2 \, dx}.$$

## Remarks

Since $T_K$ is self-adjoint, $R_K(\varphi)$ is a real number, and $R_K(c \varphi) = R_K(\varphi)$ for every nonzero $c \in \C$.
If $\varphi$ is an eigenfunction with eigenvalue $\mu$, then $R_K(\varphi) = \mu$.

Example: for $K(x, y) = xy$ on $[0, 1]$ and $\varphi = 1$, the numerator is $\int_0^1 \int_0^1 xy \, dx \, dy = 1/4$ and the denominator is $1$, so $R_K(1) = 1/4$.
For $\varphi(x) = x$, the numerator is $\left(\int_0^1 x^2 \, dx\right)^2 = 1/9$ and the denominator is $1/3$, so $R_K(\varphi) = 1/3$, the eigenvalue of $x$.

Non-example: the constant kernel $K(x, y) = i$ is not Hermitian, since $\overline{K(x, y)} = -i \ne K(y, x)$.
The same formula at $\varphi = 1$ on $[0, 1]$ gives $i$, which is not real, so it is not a Rayleigh quotient in the sense above.

For a Hermitian matrix $A$, the same formula with sums instead of integrals is the [[Rayleigh quotient|06WHP8SE98]] $x^* A x / x^* x$.