DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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Rayleigh-quotient iteration for Hermitian matrices

Definition

Let $F \in {R, C}$ , let $A \in M_{n} (F)$ be self-adjoint (Hermitian in the complex case), and let $x_{0} \in F^{n}$ have unit length in the standard inner-product norm. The Rayleigh-quotient iteration is the adaptive shifted inverse iteration defined by

$μ_{k} : = ⟨ A x_{k}, x_{k} ⟩, y_{k + 1} : = (A - μ_{k} I)^{- 1} x_{k}, x_{k + 1} : = \frac{y_{k + 1}}{∥ y_{k + 1} ∥_{2}} .$

whenever $A - μ_{k} I$ is invertible. Because $A$ is Hermitian and $x_{k}$ is unit length, $μ_{k}$ is the Rayleigh quotient $⟨ A x_{k}, x_{k} ⟩ / ⟨ x_{k}, x_{k} ⟩$ . Thus each step uses the current Rayleigh quotient as the shift.

Depends on

Used by

Near a simple Hermitian eigenvector, Rayleigh-quotient iteration converges cubically Theorem

Dependency tree · two levels

8 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

Andrew Stuart and Jochen Voss, Matrix Analysis and Algorithms (standard reference, not scraped)
Per-Olof Persson, The QR Algorithm I (standard reference, not scraped)