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.
The basis-independent characteristic polynomial of an endomorphism of a finite-dimensional space, including in dimension zero
Definition
Let be an endomorphism of a finite-dimensional vector space over . Choose an ordered basis and define
Matrices representing the same endomorphism in different ordered bases are similar, and similar matrices have the same characteristic polynomial. Hence is independent of . When , the empty ordered basis gives .
Depends on
- For $A\in M_n(F)$, the characteristic polynomial is $\chi_A(x)=\det(xI_n-A)$ when $n\geq1$, with $\chi_A(x)=1$ for the unique $0\times0$ matrix
- Similar matrices have the same characteristic polynomial
- Coordinate columns $[v]_{\mathcal B}$ and matrices $[T]_{\mathcal B}^{\mathcal C}$ of linear maps relative to ordered bases
- Similarity is an equivalence relation, and two matrices represent the same endomorphism in two bases exactly when they are similar
Used by
- Every endomorphism of a nonzero finite-dimensional vector space over an algebraically closed field has an eigenvalue Corollary
- If χ_T splits over F, every eigenvalue of χ_T(T) is 0 Corollary
- The inverse of an invertible finite-dimensional endomorphism is a polynomial in that endomorphism Corollary
- Algebraic multiplicity as the exponent of x-λ in χ_T, and geometric multiplicity as dim E_λ(T) Definition
- Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial, χ_T(T)=0 Theorem
- For every finite-dimensional space, σ_F(T) is exactly the set of roots in F of χ_T Theorem
- If χ_T(x)=∏_i<n(x-λᵢ) in F[x], then det(T)=∏_i<nλᵢ: determinant is the product of the eigenvalues counted with algebraic multiplicity Theorem
- If χ_T(x)=∏_i<n(x-λᵢ) in F[x], then tr(T)=∑_i<nλᵢ: trace is the sum of the eigenvalues counted with algebraic multiplicity 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
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 64 results over 15 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
- H. Pinkham, Linear Algebra, §§12.1–12.4 (standard reference, not scraped)
- S. Axler, Linear Algebra Done Right, 4th ed., §9 (standard reference, not scraped)