ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge 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.
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Shifted inverse iteration can target a non-dominant eigenvalue by moving it closest to the shift

Example

Let

$A = diag (5, 1), μ = 0.8, x_{0} = \frac{1}{\sqrt{2}} [\begin{array}{c} 1 \\ 1 \end{array}] .$

Then shifted inverse iteration converges to the eigendirection of the smaller eigenvalue $1$ , because that eigenvalue is nearest to the shift.

Facts & Assumptions

Given: The diagonal matrix $A$ , the shift $μ = 0.8$ , and the start vector $x_{0}$ .

[L1]

For a diagonalisable matrix, a shift outside the spectrum, a uniquely nearest simple eigenvalue, and a start vector with nonzero component in its eigendirection, shifted inverse iteration converges to that eigendirection (If $μ$ is not an eigenvalue and one simple eigenvalue is uniquely nearest to $μ$ , shifted inverse iteration converges to its eigendirection).

Verification

technique · computation

1.1

Here $(A - μ I)^{- 1} = diag ⁣ (\frac{5}{21}, 5) .$ The transformed eigenvalue magnitudes are $5 / 21$ and $5$ , so the eigendirection $e_{2}$ of the original eigenvalue $1$ becomes dominant.

algebra

2.1

The start vector has nonzero $e_{2}$ -component, so [L1] applies and the normalised iterates converge to the line $F e_{2}$ .

L1step 1.1∎

Depends on

If $\mu$ is not an eigenvalue and one simple eigenvalue is uniquely nearest to $\mu$, shifted inverse iteration converges to its eigendirection

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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