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.

Inverse iteration and shifted inverse iteration

Definition

Let $A \in M_{n} (F)$ , let $μ \in F$ with $A - μ I$ invertible, and let $x_{0} \in F^{n}$ be nonzero. The shifted inverse iteration with shift $μ$ and starting vector $x_{0}$ is the power iteration applied to $(A - μ I)^{- 1}$ :

x_{k + 1} : = \frac{(A - μ I)^{- 1} x_{k}}{∥ (A - μ I)^{- 1} x_{k} ∥_{2}} .

The special case $μ = 0$ , when $A$ itself is invertible, is inverse iteration.

Thus shifted inverse iteration is defined exactly when $A - μ I \in {GL}_{n} (F)$ in the sense of Invertible matrices and the general linear group ${GL}_{n} (F)$ .

Depends on

Used by

Dependency tree · two levels

4 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)