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
- Semisimple endomorphisms as endomorphisms diagonalisable over an algebraic closure, and nilpotent endomorphisms Definition
- 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
- FALSE: Every endomorphism has a commuting diagonal-plus-nilpotent decomposition over its base field False statement
- An endomorphism is diagonalisable exactly when V=⨁_i<rE_λᵢ(T) for some finite list of distinct scalars λᵢ Theorem
- If a diagonalisable matrix has a simple eigenvalue of strictly largest modulus and the start vector has a nonzero component in that eigendirection, power iteration converges projectively at the eigenvalue-ratio rate Theorem
- Subspace iteration converges to the dominant invariant subspace when a spectral gap separates the wanted and unwanted eigenvalues Theorem
- Under separated moduli and leading-minor hypotheses, unshifted QR drives the strict lower triangle to zero and orders the eigenvalues on the diagonal Theorem
Dependency tree · two levels
19 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
- 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)