PropositionStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-31

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An Arnoldi Ritz pair has residual norm controlled by the last Hessenberg subdiagonal entry

Statement

Assume Arnoldi runs through step $m$ without breakdown, let $H_{m} = Q_{m}^{*} A Q_{m}$ , and let $H_{m} y = θ y$ with $∥ y ∥_{2} = 1$ . If $u = Q_{m} y$ is the associated Ritz vector, then

$A u - θ u = h_{m + 1, m} (e_{m}^{*} y) v_{m + 1} .$

In particular,

$∥ A u - θ u ∥_{2} = ∣ h_{m + 1, m} ∣ ∣ e_{m}^{*} y ∣ .$

Facts & Assumptions

Given: An Arnoldi run through step $m$ without breakdown, a unit eigenvector $y$ of $H_{m}$ , and the Ritz vector $u = Q_{m} y$ .

[F1]

In Arnoldi, Ritz values are eigenvalues of $H_{m} = Q_{m}^{*} A Q_{m}$ and Ritz vectors have the form $u = Q_{m} y$ (Ritz values and Ritz vectors extracted from the Arnoldi Hessenberg reduction).

[L1]

Arnoldi gives $A Q_{m} = Q_{m + 1} {\overset{ˉ}{H}}_{m}$ , where ${\overset{ˉ}{H}}_{m} = [\begin{array}{c} H_{m} \\ h_{m + 1, m} e_{m}^{*} \end{array}]$ (Before breakdown, Arnoldi produces an orthonormal Krylov basis and a rectangular upper-Hessenberg factorization).

Proof

technique · direct

1.1

Using [F1] and [L1], $A u = A Q_{m} y = Q_{m + 1} {\overset{ˉ}{H}}_{m} y = Q_{m} H_{m} y + h_{m + 1, m} (e_{m}^{*} y) v_{m + 1} .$ Since $H_{m} y = θ y$ , this becomes $A u = θ Q_{m} y + h_{m + 1, m} (e_{m}^{*} y) v_{m + 1} .$

F1L1algebra

2.1

Because $Q_{m} y = u$ , step 1.1 is exactly $A u - θ u = h_{m + 1, m} (e_{m}^{*} y) v_{m + 1}$ . The Arnoldi vectors are orthonormal, so $∥ v_{m + 1} ∥_{2} = 1$ , and the norm formula follows.

step 1.1L1algebra∎

Depends on

Used by

A Ritz pair from Arnoldi has a residual given by the last Hessenberg column entry Example

Dependency tree · two levels

6 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

Lloyd N. Trefethen and David Bau III, Numerical Linear Algebra (standard reference, not scraped)