Alphabeta MathHow 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.
An invertible matrix can fail unpivoted LU at the first pivot
Statement refuted
Every invertible square matrix has an unpivoted unit-lower LU factorisation.
Facts & Assumptions
Given: The matrix
A square matrix has an unpivoted unit-lower LU factorisation with nonzero pivots exactly when every leading principal minor is nonzero (A square matrix has an unpivoted unit-lower LU factorisation exactly when all leading principal minors are nonzero).
Counterexample
The determinant of is , so is invertible. Its first leading principal minor is the determinant .
By [L1], the vanishing of forbids an unpivoted unit-lower LU factorisation with nonzero pivots. Therefore invertibility alone does not suffice.
Step 2.1 refutes the statement.
Depends on
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
- Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, Example 2.6.1 (standard reference, not scraped)