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.

Finite centering matrices and centered data

Definition

For an integer $n \geq 1$ , let $1_{n} \in R^{n}$ be the vector whose entries are all $1$ , and let $I_{n}$ be the $n \times n$ identity matrix. The centering matrix is

$H_{n} : = I_{n} - \frac{1}{n} 1_{n} 1_{n}^{T} .$

For $v \in R^{n}$ , write $\overset{ˉ}{v} : = n^{- 1} 1_{n}^{T} v$ ; then $H_{n} v = v - \overset{ˉ}{v} 1_{n}$ . For a finite data matrix $X \in R^{n \times p}$ with $p \geq 1$ , its centered data matrix is $X_{c} : = H_{n} X$ ; each column of $X_{c}$ has sum zero.

Definition references

Hastie, Tibshirani, and Friedman, ESL II, §14.5.1, printed pp. 534–541

Cameron, Linear and Nonlinear Dimensionality Reduction, §2.2, PDF p. 4

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

The Elements of Statistical Learning, second edition, corrected 12th printing (standard reference, not scraped)
Maria Cameron, Linear and Nonlinear Dimensionality Reduction, lecture notes (standard reference, not scraped)