ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge 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.

The unbiased sample covariance is singular when 2 ≤ n ≤ p

Example

For every dataset of $n$ vectors in $R^{p}$ with $2 \leq n \leq p$ , the covariance convention $S = R / (n - 1)$ from The iid multivariate-normal sample and covariance conventions is singular. Equality $n = p$ is included. At $n = 1$ , that covariance convention is undefined because its denominator is zero.

Facts & Assumptions

Given: Integers $2 \leq n \leq p$ and arbitrary vectors $x_{1}, \dots, x_{n} \in R^{p}$ .

[F1]

The centered scatter is $R = \sum_{i = 1}^{n} (x_{i} - \overset{ˉ}{x}) (x_{i} - \overset{ˉ}{x})^{T}$ and $S = R / (n - 1)$ . The iid multivariate-normal sample and covariance conventions

[F2]

Rank is the dimension of the image, nullity is the dimension of the kernel, and their sum equals the dimension of the finite-dimensional domain. Rank and nullity of a linear map with finite-dimensional domain Rank-nullity: $\dim_{F} V = nullity T + rank T$

Verification

1.1givenF1algebra

Let $v_{i} = x_{i} - \overset{ˉ}{x}$ . Their sum is $\sum_{i} v_{i} = 0$ , so the $n$ centered vectors are linearly dependent. Their span therefore has dimension at most $n - 1 \leq p - 1$ .

2.1F1F2step 1.1algebra

Let $V = [v_{1} ∣ \dots ∣ v_{n}] : R^{n} \to R^{p}$ . Since $V (1, \dots, 1)^{T} = 0$ , [F2] gives $rank V \leq n - 1 \leq p - 1$ , so $im V$ is a proper subspace of $R^{p}$ . The scatter matrix is $R = V V^{T}$ , hence $im R \subseteq im V$ ; it is not surjective and therefore is singular. Because $n - 1 > 0$ , $S = R / (n - 1)$ has the same non-surjective image and is singular.

3.1F1step 1.1step 2.1∎

If $n = p$ , the rank bound in step 1.1 is $p - 1$ , so equality of sample size and dimension still forces singularity. If $n = 1$ , the definition $S = R / (n - 1)$ has zero denominator and defines no sample covariance; this example makes no $n = 1$ covariance claim.

Depends on

Used by

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Sources

Arak Mathai, Serge Provost, and Hans J. Haubold, Multivariate Statistical Analysis in the Real and Complex Domains, Chapter 3 §3.5, PDF pp. 55–62 (standard reference, not scraped)
Soumendu Mukherjee, Multivariate Analysis 2021, Lecture 3, PDF pp. 1–5 (standard reference, not scraped)
Peter D. Hoff, Multivariate Statistical Analysis, Duke STA 832 lecture notes, relevant sections in Chapters 4 and 7 (standard reference, not scraped)