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.
Polynomial evaluation at an endomorphism:
Definition
Let be an endomorphism and let . Define
where and . The sum is finite because the coefficient sequence of has finite support. In particular, the zero polynomial evaluates to the zero endomorphism and the constant polynomial evaluates to .
Depends on
Used by
- Each projection in the primary decomposition is a polynomial in the endomorphism Corollary
- The algebra F[T] generated by an endomorphism is isomorphic to F[x]/(μ_T) Corollary
- The commutant of a cyclic endomorphism consists of its polynomials Corollary
- Cyclic subspaces, cyclic vectors, and vector annihilators Definition
- Primary components ker q(T)ᵉ and generalised eigenspaces G_λ⁽ᵉ⁾(T)=ker(T-λ I)ᵉ Definition
- The annihilator set Ann(T)={p∈ F[x]:p(T)=0}; once existence is proved, its unique monic generator μ_T is the minimal polynomial Definition
- The F[x]-module V_T of an endomorphism Definition
- For every polynomial p, both ker p(T) and imp(T) are T-invariant Lemma
- If gcd(f,g)=1 and (fg)(T)=0, then V=ker f(T)⊕ker g(T) Lemma
- Polynomial evaluation commutes with restriction and invariant quotients Proposition
- An endomorphism is diagonalisable if and only if its minimal polynomial is a product of distinct linear factors Theorem
- Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial, χ_T(T)=0 Theorem
- Every finite cyclic extension has a normal basis Theorem
- If χ_T(x)=∏_i<n(x-λᵢ) in F[x], then χ_p(T)(y)=∏_i<n(y-p(λᵢ)) for every p∈ F[x]: the eigenvalues of p(T) are p(λᵢ), counted with algebraic multiplicity Theorem
- Over every extension field, a scalar is an eigenvalue of the extended matrix exactly when it is a root of the minimal polynomial Theorem
- The minimal and characteristic polynomials have exactly the same monic irreducible factors Theorem
Dependency tree · two levels
7 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
- M. Khovanov, Linear Algebra II notes, §6 (standard reference, not scraped)
- H. Pinkham, Linear Algebra, §12.3 (standard reference, not scraped)