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.

Unshifted QR is orthonormalised simultaneous iteration, and every QR iterate is unitarily similar to the original matrix

Statement

Let $(A_{k})$ be the unshifted QR iteration started from $A = A_{0}$ , and put ${\hat{Q}}_{0} : = I$ and ${\hat{Q}}_{k} : = Q_{0} Q_{1} \dots Q_{k - 1}$ for $k \geq 1$ .

$A_{k} = {\hat{Q}}_{k}^{*} A {\hat{Q}}_{k}$ for every $k$ .
The columns of ${\hat{Q}}_{k}$ are obtained by orthonormalising the columns of $A^{k}$ , so unshifted QR is simultaneous iteration followed by orthonormalisation.

Facts & Assumptions

Given: An unshifted QR iteration $A_{k} = Q_{k} R_{k}$ , $A_{k + 1} = R_{k} Q_{k}$ .

[L1]

Unshifted QR iteration is defined by the factorisation and update above (Unshifted QR iteration, shifted QR iteration, deflation, and the Wilkinson shift).

[L2]

Subspace iteration is repeated QR orthonormalisation of powers of $A$ (Subspace iteration and the dominant invariant subspace of a matrix).

Proof

technique · direct

1.1

From [L1], $A_{k + 1} = R_{k} Q_{k} = Q_{k}^{*} A_{k} Q_{k} .$ Iterating this identity yields $A_{k} = {\hat{Q}}_{k}^{*} A {\hat{Q}}_{k}$ .

L1algebra

1.2

Also $A_{k} Q_{k} = Q_{k} R_{k} Q_{k} = Q_{k} A_{k + 1}$ , so multiplying the factorisations gives $A^{k} = (Q_{0} Q_{1} \dots Q_{k - 1}) (R_{k - 1} \dots R_{1} R_{0}) = {\hat{Q}}_{k} {\hat{R}}_{k}$ with ${\hat{R}}_{k}$ upper triangular. Thus the columns of ${\hat{Q}}_{k}$ are the orthonormalised columns of $A^{k}$ .

L1algebra

2.1

Step 1.1 proves unitary similarity, and step 1.2 identifies the iteration with simultaneous orthonormalised power iteration as in [L2].

L2step 1.1step 1.2∎

Depends on

Used by

Under separated moduli and leading-minor hypotheses, unshifted QR drives the strict lower triangle to zero and orders the eigenvalues on the diagonal Theorem

Dependency tree · two levels

5 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

Per-Olof Persson, The QR Algorithm I (standard reference, not scraped)