ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 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.

A Wilkinson-shifted QR step can sharply reduce the tail of a symmetric tridiagonal matrix

Example

Let

$T = [\begin{array}{ccc} 4 & 1 & 0 \\ 1 & 3 & 0.1 \\ 0 & 0.1 & 2 \end{array}] .$

The trailing $2 \times 2$ block has eigenvalues about $3.0099$ and $1.9901$ , so the Wilkinson shift is $μ \approx 1.9901$ . One shifted QR step gives

$T_{+} \approx [\begin{array}{ccc} 4.5992 & 0.2091 & 0 \\ 0.2091 & 2.4199 & - 0.0020 \\ 0 & - 0.0020 & 1.9809 \end{array}] .$

Thus the last subdiagonal entry shrinks from $0.1$ to about $0.0020$ .

Facts & Assumptions

Given: The symmetric tridiagonal matrix $T$ of the statement.

[L1]

The Wilkinson shift is the eigenvalue of the trailing $2 \times 2$ block that is nearer to the bottom-right entry, away from ties (Unshifted QR iteration, shifted QR iteration, deflation, and the Wilkinson shift).

Verification

technique · computation

1.1

The trailing $2 \times 2$ block of $T$ is $[\begin{matrix} 3 & 1 / 10 \\ 1 / 10 & 2 \end{matrix}]$ , whose eigenvalues are $(25 \pm \sqrt{26}) / 10$ . The smaller one is nearer to the bottom-right entry $2$ , so it is the Wilkinson shift.

algebra

2.1

Carrying out one shifted QR step with that shift gives the displayed matrix $T_{+}$ , whose bottom subdiagonal magnitude is about $2 \times 10^{- 3}$ , hence smaller than $1 / 400$ and far below the original $10^{- 1}$ . This is a direct one-step computation using the shift from [L1].

L1step 1.1algebra∎

Depends on

Unshifted QR iteration, shifted QR iteration, deflation, and the Wilkinson shift

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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.