PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

Shifted QR iteration preserves upper Hessenberg form

Statement

If $H$ is upper Hessenberg and a shifted QR step factors $H - μ I = Q R$ using the standard adjacent-row Givens eliminations, then

$H_{+} : = R Q + μ I$

is again upper Hessenberg.

Facts & Assumptions

Given: An upper Hessenberg matrix $H$ and a shifted QR factorisation $H - μ I = Q R$ built from adjacent Givens eliminations.

[L1]

A shifted QR step is $H_{+} = R Q + μ I$ (Unshifted QR iteration, shifted QR iteration, deflation, and the Wilkinson shift).

[L2]

Upper Hessenberg means all entries below the first subdiagonal vanish (Upper Hessenberg matrices and real symmetric tridiagonal matrices).

[L3]

Givens transformations are unitary and annihilate a chosen adjacent entry (Real Givens rotations and complex Givens transformations, Householder reflectors and Givens transformations are unitary and can annihilate prescribed entries).

Proof

technique · direct

1.1L2L3construct

Because $H - μ I$ is upper Hessenberg, each subdiagonal entry can be annihilated by an adjacent Givens rotation acting only on two consecutive rows. By [L3], the product of these rotations gives $Q$ and an upper triangular $R$ .

2.1step 1.1algebra

Right-multiplying an upper triangular matrix by one adjacent Givens rotation can create a nonzero entry only one row below the diagonal in the two affected columns. Repeating this through the same adjacent sequence keeps $R Q$ upper Hessenberg.

3.1L1step 2.1∎

Adding $μ I$ changes only diagonal entries, so [L1] and step 2.1 show that $H_{+} = R Q + μ I$ is again upper Hessenberg.

Depends on

Used by

A real Wilkinson-shifted symmetric tridiagonal QR step preserves symmetric tridiagonal form away from ties Theorem

Dependency tree · two levels

10 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)