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.
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.

One unshifted QR step and one shifted QR step on the same symmetric matrix show the role of the shift

Example

Let

$A = [\begin{array}{cc} 3 & 1 \\ 1 & 2 \end{array}] .$

One unshifted QR step produces

$A_{1} = [\begin{array}{cc} \frac{7}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{3}{2} \end{array}],$

while a shifted QR step with $μ = \frac{3}{2}$ produces

${\tilde{A}}_{1} = [\begin{array}{cc} \frac{47}{13} & - \frac{1}{13} \\ - \frac{1}{13} & \frac{18}{13} \end{array}] .$

The shifted step reduces the off-diagonal entry from $1 / 2$ to $1 / 13$ .

Facts & Assumptions

Given: The symmetric matrix $A$ and the shift $μ = 3 / 2$ .

[L1]

Unshifted and shifted QR steps are defined by the factorisations $A = Q R$ , $A_{1} = R Q$ and $(A - μ I) = Q R$ , ${\tilde{A}}_{1} = R Q + μ I$ (Unshifted QR iteration, shifted QR iteration, deflation, and the Wilkinson shift).

Verification

technique · computation

1.1

For the unshifted step, a QR factorisation of $A$ gives $Q = \frac{1}{\sqrt{10}} [\begin{array}{cc} 3 & - 1 \\ 1 & 3 \end{array}], R = \frac{1}{\sqrt{10}} [\begin{array}{cc} 10 & 5 \\ 0 & 5 \end{array}],$ so [L1] yields $A_{1} = R Q = [\begin{matrix} 7 / 2 & 1 / 2 \\ 1 / 2 & 3 / 2 \end{matrix}]$ .

L1algebra

1.2

For the shifted step, $A - \frac{3}{2} I = [\begin{array}{cc} \frac{3}{2} & 1 \\ 1 & \frac{1}{2} \end{array}],$ and a QR factorisation gives $Q = \frac{1}{\sqrt{13}} [\begin{array}{cc} 3 & - 2 \\ 2 & 3 \end{array}] .$ Using [L1] again gives ${\tilde{A}}_{1} = [\begin{matrix} 47 / 13 & - 1 / 13 \\ - 1 / 13 & 18 / 13 \end{matrix}]$ .

L1algebra

2.1

Comparing steps 1.1 and 1.2 shows the effect of the shift: the off-diagonal magnitude drops from $1 / 2$ to $1 / 13$ in a single step.

step 1.1step 1.2∎

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.