LemmaStatement: AI-generatedProof: AI-generated

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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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A negative adjacency eigenvalue survives all-ones centering

Statement

For a real symmetric $n \times n$ matrix $B$ , $n \geq 1$ , and $p \geq 0$ , let $λ_{\min} (B)$ be its least eigenvalue and let $∥ \cdot ∥$ be the vertex Hilbert-space operator norm. With $J$ the all-ones matrix, $∥ B - p J ∥ \geq - λ_{\min} (B)$ .

Statement references

Finite vertex Hilbert space in the article convention

Self-adjoint order, norm bounds and diagonal tests

Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely

The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖

Facts & Assumptions

[F1]

In the finite-dimensional complex vertex space, a real symmetric matrix is self-adjoint and therefore normal. A normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis; the vertex convention has the same adjoint and orthogonality as the library convention. Finite vertex Hilbert space in the article convention Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely

[F2]

The induced operator norm is $∥ M ∥ = \max_{∥ v ∥_{2} = 1} ∥ M v ∥_{2}$ on the nonzero vertex space. For a real symmetric matrix $M$ , the norm-order equivalence gives $- ∥ M ∥ I ⪯ M ⪯ ∥ M ∥ I$ , hence $∣ u^{*} M u ∣ \leq ∥ M ∥$ for every unit vector $u$ . The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖ Self-adjoint order, norm bounds and diagonal tests

Proof

Given: The data and hypotheses in the Statement.

Goal: Establish the stated conclusions.

1.1F1givenalgebra

By [F1], $B$ has an orthonormal eigenbasis. Each eigenvalue is real because, for a unit eigenvector $v$ , it equals the real quadratic value $v^{*} B v$ . Since $n \geq 1$ , choose a unit eigenvector $u$ for the least eigenvalue $λ_{\min} (B)$ among this finite nonempty spectrum. Since $u^{*} J u = ∣ \sum_{i} u_{i} ∣^{2} \geq 0$ , one has $u^{*} (B - p J) u = λ_{\min} (B) - p ∣ \sum_{i} u_{i} ∣^{2} \leq λ_{\min} (B)$ .

2.1F2step 1.1algebra∎

The matrix $B - p J$ is real symmetric, so the norm bound in [F2] gives $∥ B - p J ∥ \geq - u^{*} (B - p J) u \geq - λ_{\min} (B)$ , proving the claim even if the right-hand side is negative.

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Used by · 2

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