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PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31
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  • 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 QR iteration preserves upper Hessenberg form

Statement

If H is upper Hessenberg and a shifted QR step factors H−μI=QR using the standard adjacent-row Givens eliminations, then

H+:=RQ+μI

is again upper Hessenberg.

Facts & Assumptions

Given: An upper Hessenberg matrix H and a shifted QR factorisation H−μI=QR built from adjacent Givens eliminations.

[L1]

A shifted QR step is H+=RQ+μI (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
1.1

Because H−μI 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.

L2L3construct
2.1

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 RQ upper Hessenberg.

step 1.1algebra
3.1

Adding μI changes only diagonal entries, so [L1] and step 2.1 show that H+=RQ+μI is again upper Hessenberg.

L1step 2.1∎

Depends on

Used by

Dependency tree · two levels

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Sources