Rayleigh quotient for symmetric kernels

Builds on Square-integrable kernel Hermitian kernel Hilbert–Schmidt integral operator Rayleigh quotient

Definition

Let $a < b$ and let $K \in L^2([a, b] \times [a, b])$ be a square-integrable kernel that is Hermitian: $K(y, x) = \overline{K(x, y)}$ for almost all $x, y$ . Let $T_K$ be its Hilbert–Schmidt integral operator 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 $x^* A x / x^* x$ .

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Hermitian kernel concept
Hilbert–Schmidt integral operator concept
Rayleigh quotient concept
Square-integrable kernel concept
Rayleigh quotient for symmetric kernels definition

Tag	`J858GV245K`
URI	`m2://J858GV245K`
Kind	definition
Writing	W0 stub
MSC	45C05, 47A75
Defines	Rayleigh quotient for symmetric kernels
Uses	Square-integrable kernel, Hermitian kernel, Hilbert–Schmidt integral operator, Rayleigh quotient

Rayleigh quotient for symmetric kernels

Definition

Remarks

Reading path

Further reading

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