DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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The Rayleigh quotient of a nonzero vector for a self-adjoint endomorphism

Definition

Let $V$ be a finite-dimensional real inner product space, let $T : V \to V$ be self-adjoint, and let $v \in V$ be nonzero. The Rayleigh quotient of $v$ for $T$ is

$R_{T} (v) : = \frac{⟨ T v, v ⟩}{⟨ v, v ⟩} .$

Because $⟨ v, v ⟩ > 0$ for nonzero $v$ in an inner product space, this quotient is a well-defined real number.

Depends on

Used by

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

MIT 18.409, Lecture 3: Courant-Fischer and Rayleigh quotients (standard reference, not scraped)