TheoremStatement: AI-adaptedProof: 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.

Certified inertia counts maintain ordered eigenvalue brackets

Statement

Let $n \geq 1$ , let $A$ be a nonempty family of Hermitian $n \times n$ matrices, and let a closed interval $[L, U]$ initially contain every ordered eigenvalue $λ_{k} (A)$ for every $A \in A$ , with eigenvalues listed increasingly. For a query shift $τ$ , suppose the query either returns inconclusive or returns an integer $ν (τ)$ such that every $A \in A$ is nonsingular at $A - τ I$ and exactly $ν (τ)$ of its eigenvalues lie below $τ$ . For each $k \in {1, \dots, n}$ , start with $[L, U]$ and repeat at most $N \in N_{0}$ times: query its midpoint $τ$ ; if $ν (τ) \geq k$ , replace $U$ by $τ$ , and otherwise replace $L$ by $τ$ ; stop when width is at most $ε > 0$ . An inconclusive query returns the current interval without changing it. Every returned interval contains $λ_{k} (A)$ for every $A \in A$ . After $s$ successful queries its width is $(U - L) 2^{- s}$ .

Facts & Assumptions

Given: The initial uniform bracket, tolerance $ε > 0$ , budget $N$ , and the stated query semantics.

[F1]

Hermitian matrices have real ordered eigenvalues (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

[A1]

Every successful query count equals the number of eigenvalues strictly below that shift for every admitted matrix; a successful query excludes a zero eigenvalue at the shift.

Proof

1.1givenF1A1algebra

Fix $k$ . The initial interval contains $λ_{k} (A)$ for every $A$ by hypothesis. At a successful shift $τ$ , if $ν (τ) \geq k$ , then at least $k$ eigenvalues of every $A$ are below $τ$ , so $λ_{k} (A) < τ$ and replacing the upper endpoint by $τ$ preserves every enclosure. If $ν (τ) < k$ , then fewer than $k$ eigenvalues of every $A$ are below $τ$ ; nonsingularity at the shift makes all remaining eigenvalues strictly above $τ$ , so $λ_{k} (A) > τ$ and replacing the lower endpoint preserves every enclosure.

2.1givenstep 1.1algebra∎

Each successful update replaces one endpoint by the midpoint and halves the width. An inconclusive query leaves the already valid interval unchanged. Induction on the number $s$ of successful queries gives width $(U - L) 2^{- s}$ and retains the eigenvalue enclosure in every return branch.

Depends on

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

Used by

Dependency tree · two levels

6 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

Nicholas J. Higham, Numerical Linear Algebra and Matrix Analysis, MIMS EPrint 2015.103, §5.6 (standard reference, not scraped)