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.
Depends on
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- Coordinate columns $[v]_{\mathcal B}$ and matrices $[T]_{\mathcal B}^{\mathcal C}$ of linear maps relative to ordered bases
- Eigenvalues, eigenvectors, eigenspaces $E_\lambda(T)=\ker(T-\lambda I)$, and the spectrum $\sigma_F(T)$ of an endomorphism
Used by
- Simultaneous diagonalisability: one basis that diagonalises every endomorphism in a family Definition
- FALSE: A diagonalisable endomorphism must have a characteristic polynomial with distinct roots False statement
- An endomorphism is diagonalisable exactly when V=bigoplus_i<rE_λᵢ(T) for some finite list of distinct scalars λᵢ Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 55 results over 19 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Sheldon Axler, Linear Algebra Done Right, 4th ed., §§5A–5B (standard reference, not scraped)
- Keith Conrad, The Minimal Polynomial and Some Applications, §§4–5 (standard reference, not scraped)