Alphabeta Math

Eigenvalues, Eigenvectors and the Characteristic Polynomial

About this page
How 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.

✓ 17 results · all verified · 8 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 9 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Coxeter Groups
Linear Algebra
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Definition

Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism

Proposition

For a finite-dimensional space, λ is an eigenvalue of T if and only if T-λ I is not invertible

Theorem

Eigenvectors belonging to pairwise distinct eigenvalues are linearly independent

Corollary

An endomorphism of an n-dimensional space has at most n distinct eigenvalues

Definition

For A∈ Mₙ(F), the characteristic polynomial is χ_A(x)=det(xIₙ-A) when n≥1, with χ_A(x)=1 for the unique 0×0 matrix

Lemma

χ_A(x) is monic of degree n; for n≥1 its xⁿ⁻¹ coefficient is -tr(A) and its constant coefficient is (-1)ⁿ det(A), while χ_0×0=1

Lemma

The characteristic polynomial of a block upper- or lower-triangular matrix is the product of the characteristic polynomials of its diagonal blocks

Theorem

Similar matrices have the same characteristic polynomial

Definition

The basis-independent characteristic polynomial χ_T of an endomorphism of a finite-dimensional space, including χ_T=1 in dimension zero

Theorem

For every finite-dimensional space, σ_F(T) is exactly the set of roots in F of χ_T

Definition

An algebraically closed field: every nonconstant polynomial has a root in the field

Corollary

Every endomorphism of a nonzero finite-dimensional vector space over an algebraically closed field has an eigenvalue

Definition

Algebraic multiplicity as the exponent of x-λ in χ_T, and geometric multiplicity as dim E_λ(T)

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 det(T)=∏_i<nλᵢ: determinant is the product of the eigenvalues counted with algebraic multiplicity

Theorem

The geometric multiplicity of an eigenvalue does not exceed its algebraic multiplicity

Corollary

An eigenvalue of algebraic multiplicity one has a one-dimensional eigenspace

Theorem

For A,B∈ Mₙ(F), the products AB and BA have the same characteristic polynomial

Definition

Polynomial evaluation at an endomorphism: p(T)=∑ₖ aₖTᵏ

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

Corollary

If χ_T splits over F, every eigenvalue of χ_T(T) is 0

Theorem

Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial, χ_T(T)=0

Corollary

The inverse of an invertible finite-dimensional endomorphism is a polynomial in that endomorphism

Sources