DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Row leverage scores of a finite-dimensional subspace

Definition

Let $K \in {R, C}$ , and let $U \subseteq K^{n}$ have dimension $d \geq 1$ . Choose a matrix $Q \in K^{n \times d}$ whose columns are an orthonormal basis of $U$ , so $Q^{*} Q = I_{d}$ . For $0 \leq i < n$ , define the $i$ -th row leverage score of $U$ by

ℓ_{i} (U) : = ∥ e_{i}^{*} Q ∥_{2}^{2},

where $e_{i}$ is the $i$ -th standard coordinate vector. The following item proves that this value is independent of the chosen orthonormal basis, so the definition is well-defined.

Remarks

Such an orthonormal basis exists for every $U$ here by Every finite-dimensional real or complex inner product space has an orthonormal basis.

Depends on

Every finite-dimensional real or complex inner product space has an orthonormal basis

Used by

Dependency tree · two levels

3 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

David P. Woodruff, Sketching as a Tool for Numerical Linear Algebra, §2.4, Definition 16 (standard reference, not scraped)
Michael W. Mahoney, Randomized Algorithms for Matrices and Data, §4.1 (standard reference, not scraped)