LemmaStatement: Literature-sourcedProof: 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 REML error-contrast covariance is positive definite

Statement

Let $X \in R^{n \times p}$ have full column rank with $p < n$ , let $K \in R^{n \times (n - p)}$ be an error-contrast matrix for $X$ , and let $V \in R^{n \times n}$ be symmetric positive definite. Then $K^{T} V K$ is symmetric positive definite. No Gaussian assumption is needed for this matrix claim.

Facts & Assumptions

Given: $V \in R^{n \times n}$ is symmetric positive definite and $K \in R^{n \times (n - p)}$ has full column rank.

[F1]

An error-contrast matrix has full column rank and columns spanning $col (X)^{⊥}$ . Error-contrast matrix for a full-rank fixed design.

[A1]

The matrix $V$ is symmetric positive definite.

Proof

Given: The full-rank design, its error-contrast matrix, and a symmetric positive-definite $V$ .

1.1A1algebra

The matrix $K^{T} V K$ is symmetric because $V$ is symmetric.

1.2F1A1algebra

Let $a \in R^{n - p}$ be nonzero. By [F1], $K$ has full column rank, so $K a \neq 0$ . Therefore $a^{T} K^{T} V K a = (K a)^{T} V (K a) > 0$ by positive definiteness of $V$ . Thus $K^{T} V K$ is positive definite.

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 prove symmetry and strict positivity of the quadratic form for every nonzero $a$ , which is exactly positive definiteness of $K^{T} V K$ .

Depends on

Error-contrast matrix for a full-rank fixed design

Used by

Restricted maximum likelihood from error contrasts Definition

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

Fan Li, STA610 Lecture 4: Maximum Likelihood Estimation and Inference for the Hierarchical Normal Model, PDF pp. 4–5 (standard reference, not scraped)
Duchateau, Janssen, and Rowlands, Linear Mixed Models, Chapter 2 §2.4.2 (standard reference, not scraped)