DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-08-30

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.

Real Givens rotations and complex Givens transformations

Definition

Let $n \geq 2$ and let $0 \leq i < j \leq n - 1$ .

A real Givens rotation in the $(i, j)$ -plane is the identity matrix except on rows and columns $i, j$ , where it has the $2 \times 2$ block

$(\begin{array}{cc} c & s \\ - s & c \end{array}) (c, s \in R, c^{2} + s^{2} = 1) .$ A complex Givens transformation in the $(i, j)$ -plane is the identity except on rows and columns $i, j$ , where it has block $(\begin{array}{cc} c & s \\ - \overline{s} & c \end{array}),$

with $c \in R_{\geq 0}$ , $s \in C$ , and $c^{2} + ∣ s ∣^{2} = 1$ . The modulus $∣ s ∣$ is the one from Real and imaginary parts, complex conjugation, and modulus.

Such transformations act only on the two chosen coordinates and are designed to zero one targeted entry while preserving the Euclidean norm from The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product.

Depends on

Used by

Dependency tree · two levels

7 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, Section 3.4.2 (standard reference, not scraped)
David Bindel, CS 4220: Numerical Analysis, Least squares and QR (standard reference, not scraped)