ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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A short dense matrix admits a worked Householder QR factorisation

Example

For $A = (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array}),$

the unit vector $v = \frac{1}{\sqrt{2 (2 - \sqrt{2})}} (\binom{1 - \sqrt{2}}{1})$ gives the Householder reflector

$H = I - 2 v v^{T} = \frac{1}{\sqrt{2}} (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array}) .$

Hence

$A = Q R, Q = H, R = H A = (\begin{array}{cc} \sqrt{2} & 0 \\ 0 & \sqrt{2} \end{array}) .$

Facts & Assumptions

Given: The displayed matrix $A$ , vector $v$ , and reflector $H$ .

[L1]

Successive Householder reflectors produce QR factorisations (Successive Householder or Givens transformations produce full and reduced QR factorisations with the standard dense operation counts).

Verification

technique · direct

1.1givenalgebra

The vector $v$ has norm $1$ , and direct multiplication of $I - 2 v v^{T}$ gives the displayed matrix $H$ . Since $H^{T} H = I$ , it is an orthogonal reflector.

2.1step 1.1L1algebra

Multiplying by $A$ gives $H A = \frac{1}{\sqrt{2}} (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array}) (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array}) = (\begin{array}{cc} \sqrt{2} & 0 \\ 0 & \sqrt{2} \end{array}) .$ Thus $A = Q R$ with $Q = H$ and $R$ upper triangular, exactly as predicted by [L1].

3.1step 1.1step 2.1∎

Steps 1.1-2.1 verify the example.

Depends on

Successive Householder or Givens transformations produce full and reduced QR factorisations with the standard dense operation counts

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, Example 3.4.1 (standard reference, not scraped)