ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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.

A nonsingular indefinite matrix with no admissible one-by-one diagonal pivot

Example

Take $A = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}) .$ This Hermitian matrix is nonsingular and indefinite. Every one-by-one diagonal pivot test fails, while the declared two-by-two pivot passes and gives an exact block $L D L^{*}$ factorization.

Facts & Assumptions

Given: The matrix $A$ above, threshold $θ \in (0, 1]$ , identity permutation, and the single declared pivot block of size two.

[F1]

Sparse symmetric-indefinite block LDL* with threshold tests and failure status: A one-by-one pivot must be nonzero and pass the stated column threshold; a two-by-two pivot must be nonsingular and pass the componentwise inverse-block test.

Verification

1.1givenalgebra

The determinant of $A$ is $- 1$ , so $A$ is nonsingular. Its eigenvectors $(1, 1)^{T}$ and $(1, - 1)^{T}$ have eigenvalues $1$ and $- 1$ , respectively; therefore $A$ is indefinite.

1.2givenF1algebra

Both diagonal candidates have pivot $0$ , while the other entry in each column has modulus $1$ . Since $θ^{- 1} ∣ 0 ∣ = 0$ , neither one-by-one test in [F1] passes.

1.3givenF1algebra

For the declared block $D = A$ , $\det D = - 1 \neq 0$ and $D^{- 1} = A$ . There are no external indices, so $c_{1} = c_{2} = 0$ and the two-by-two threshold left side is zero; the test in [F1] passes.

2.1step 1.3algebra∎

The algorithm therefore returns $P = I$ , $L = I$ , and $D_{b l k} = A$ . Direct multiplication gives $A = L D_{b l k} L^{*}$ . Thus the block pivot succeeds where every one-by-one diagonal pivot fails.

Depends on

Sparse symmetric-indefinite block LDL* with threshold tests and failure status

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Jennifer Scott and Miroslav Tůma, Algorithms for Sparse Linear Systems, Chapter 7, §§7.2.4 and 7.3 (standard reference, not scraped)
Timothy A. Davis, Siva Rajamanickam, and Wissam M. Sid-Lakhdar, A survey of direct methods for sparse linear systems, §11 (standard reference, not scraped)