DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-09-01

How statement and proof provenance work

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The Lanczos process as Arnoldi specialized to a Hermitian matrix, with the usual real-subdiagonal phase convention

Definition

Let $A$ be Hermitian and let $q_{1} = b / ∥ b ∥_{2}$ for a nonzero start vector $b$ . Running Arnoldi on $(A, b)$ is called the Lanczos process when one records the coefficients, at every step before Arnoldi breakdown, in the three-term form

$A q_{j} = β_{j - 1} q_{j - 1} + α_{j} q_{j} + β_{j} q_{j + 1} (j \geq 1, q_{0} : = 0, β_{0} : = 0),$

with $α_{j} \in R$ and the phase of $q_{j + 1}$ chosen so that $β_{j} \geq 0$ is real. In matrix form this is the Hermitian specialization of The Arnoldi process for building an orthonormal basis of a Krylov subspace.

Depends on

Used by

With a Hermitian matrix, Arnoldi collapses to the Lanczos three-term recurrence and a real symmetric tridiagonal projection Theorem

Dependency tree · two levels

7 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

Richard Barrett et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (standard reference, not scraped)