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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The residual $r = A x - μ x$ and the normwise backward error of an approximate eigenpair

Definition

Let $F \in {R, C}$ , let $A \in M_{n} (F)$ , let $μ \in F$ , and let $x \in F^{n}$ be nonzero. Equip $F^{n}$ with its standard inner product and Euclidean norm $∥ \cdot ∥_{2}$ . For a matrix $E \in M_{n} (F)$ , write

$∥ E ∥_{2} : = \sup_{∥ y ∥_{2} = 1} ∥ E y ∥_{2}$

for the induced operator norm of The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||. The eigenpair residual of $(μ, x)$ for $A$ is

$r (A, μ, x) : = A x - μ x .$

If $∥ x ∥_{2} = 1$ , the normwise backward error of the approximate eigenpair $(μ, x)$ is

$η (A, μ, x) : = \inf {∥ E ∥_{2} : (A + E) x = μ x} .$

Thus $η (A, μ, x)$ measures the smallest spectral-norm perturbation that makes $(μ, x)$ an exact eigenpair.

Depends on

Used by

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

Netlib Templates, Numerical Stability and Conditioning (standard reference, not scraped)

The residual r=Ax−μxr=Ax−μx and the normwise backward error of an approximate eigenpair