ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-30

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Givens QR can zero selected entries of a sparse matrix one at a time

Example

For $A = (\begin{array}{cc} 3 & 4 \\ 4 & 3 \\ 0 & 2 \end{array}),$ first apply a Givens rotation in rows $1, 2$ with $c = 3 / 5$ , $s = 4 / 5$ , then a second Givens rotation in rows $2, 3$ with $c = - 7 / \sqrt{149}$ , $s = 10 / \sqrt{149}$ . The product zeros one selected entry at a time and yields $R = (\begin{array}{cc} 5 & \frac{24}{5} \\ 0 & \frac{\sqrt{149}}{5} \\ 0 & 0 \end{array}) .$

Facts & Assumptions

Given: The displayed matrix $A$ and the two named Givens rotations.

[L1]

Givens transformations are unitary and can annihilate a chosen second coordinate (Householder reflectors and Givens transformations are unitary and can annihilate prescribed entries).

[L2]

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

Verification

technique · direct

1.1L1algebra

The first rotation $G_{1} = (\begin{array}{ccc} 3 / 5 & 4 / 5 & 0 \\ - 4 / 5 & 3 / 5 & 0 \\ 0 & 0 & 1 \end{array})$ sends the first column $(3, 4, 0)^{T}$ to $(5, 0, 0)^{T}$ , so it zeros the $(2, 1)$ entry while preserving the zero in position $(3, 1)$ . Applied to the second column, it gives $(24 / 5, - 7 / 5, 2)^{T}$ .

2.1step 1.1L1algebra

The second rotation $G_{2} = (\begin{array}{ccc} 1 & 0 & 0 \\ 0 & - 7 / \sqrt{149} & 10 / \sqrt{149} \\ 0 & - 10 / \sqrt{149} & - 7 / \sqrt{149} \end{array})$ acts only on rows $2, 3$ , so it keeps the first column fixed and sends $(- 7 / 5, 2)^{T}$ to $(\sqrt{149} / 5, 0)^{T}$ . Hence $R = G_{2} G_{1} A = (\begin{array}{cc} 5 & 24 / 5 \\ 0 & \sqrt{149} / 5 \\ 0 & 0 \end{array}),$ the displayed upper-triangular matrix.

3.1step 2.1L2algebra

Since both rotations are unitary, $Q = (G_{2} G_{1})^{*}$ and $A = Q R$ . This verifies the Givens QR factorisation promised by [L2].

4.1step 1.1step 2.1step 3.1∎

Steps 1.1-3.1 verify the example.

Depends on

Used by

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Sources

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, Givens rotations section (standard reference, not scraped)