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.
Jordan bases and Jordan canonical forms over the base field
Definition
A Jordan basis over for an endomorphism is an ordered basis obtained by concatenating Jordan strings for , all with eigenvalues in (Jordan blocks, Jordan strings, and their endpoints). The matrix of in such a basis is block diagonal with blocks and is called a Jordan canonical form of over .
The adjective “canonical” records the block multiset, not a prescribed order of blocks: permuting the strings usually changes the literal block diagonal matrix. On the zero space the empty basis and empty block matrix form the Jordan basis and Jordan form.