ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-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 regular pencil and an identically singular determinant pencil

Example

The pair $(I_{2}, 0)$ is regular and has only infinite generalized eigenvalues. The pair $(diag (1, 0), 0)$ has an identically zero determinant pencil and is singular. Thus a singular second matrix is compatible with regularity, while the determinant polynomial decides pencil regularity.

Facts & Assumptions

Given: The regularity definition for finite square pencils in Regular matrix pencils and homogeneous generalized eigenvalues.

[F1]

Regular matrix pencils and homogeneous generalized eigenvalues: a pencil is regular exactly when $\det (A - λ B)$ is not the zero polynomial.

Verification

1.1givenF1algebra

For $(A, B) = (I_{2}, 0)$ , $\det (A - λ B) = \det I_{2} = 1$ , so the pencil is regular. Its homogeneous determinant is $\det (β I_{2}) = β^{2}$ , which vanishes only at $[1 : 0]$ , the infinite eigenvalue pair.

2.1F1step 1.1algebra∎

For $(A, B) = (diag (1, 0), 0)$ , $\det (A - λ B) = \det A = 0$ for every $λ$ , so the determinant polynomial is identically zero and the pencil is singular.

Depends on

Regular matrix pencils and homogeneous generalized eigenvalues

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

Daniel Kressner, Numerical Methods and Software for General and Structured Eigenvalue Problems (PhD dissertation, 2004), Chapter 2: The QZ Algorithm (standard reference, not scraped)
LAPACK Users' Guide, Generalized Nonsymmetric Eigenproblems (standard reference, not scraped)