LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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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.

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Positive Gaussian observation noise makes the observed covariance positive definite

Statement

Let $n \geq 1$ , let $K \in R^{n \times n}$ be symmetric positive semidefinite, let $σ^{2} \geq 0$ , and put $C = K + σ^{2} I$ . If $σ^{2} > 0$ , then $C$ is symmetric positive definite, even when $K$ is singular. If $σ^{2} = 0$ , then $C$ is positive definite exactly when $K$ is positive definite; zero noise is not guaranteed to make the solve available.

Facts & Assumptions

Given: A symmetric matrix $K$ with $v^{T} K v \geq 0$ for all $v \in R^{n}$ , and a real $σ^{2} \geq 0$ .

[F1]

A real matrix is positive definite when it is symmetric and $v^{T} A v > 0$ for every nonzero $v$ , as in Hermitian positive-definite matrices and Cholesky factorisation A = LL* with positive diagonal.

Proof

1.1givenF1algebra

The matrix $C$ is symmetric because $K$ and $I$ are symmetric. For every nonzero $v$ , $v^{T} C v = v^{T} K v + σ^{2} ∥ v ∥_{2}^{2}$ . If $σ^{2} > 0$ , the first term is nonnegative and the second is strictly positive, so $v^{T} C v > 0$ ; hence $C$ is positive definite by [F1].

2.1givenF1algebra∎

If $σ^{2} = 0$ , then $C = K$ , so $C$ is positive definite if and only if $K$ is. This permits zero-noise solves exactly in the positive-definite observed-Gram cases and makes no pseudoinverse assertion in the singular cases.

Depends on

Hermitian positive-definite matrices and Cholesky factorisation A = LL* with positive diagonal

Used by

Dependency tree · two levels

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Sources

Carl Edward Rasmussen and Christopher K. I. Williams, Gaussian Processes for Machine Learning, Chapter 2 (standard reference, not scraped)
Niranjan Srinivas et al., Gaussian Process Optimization in the Bandit Setting, §2 (standard reference, not scraped)