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.
A diagonalisable endomorphism is one admitting a basis of eigenvectors, equivalently a diagonal matrix representation
Definition
Let be an endomorphism of a finite-dimensional vector space over a field . The endomorphism is diagonalisable over if has a basis consisting of eigenvectors of (Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism, Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis).
Equivalently, is diagonalisable if some ordered basis makes its matrix diagonal (Coordinate columns and matrices of linear maps relative to ordered bases). The unique endomorphism of the zero space is diagonalisable: its empty ordered basis is a basis of eigenvectors and its matrix is diagonal.