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Algebraic multiplicity as the exponent of in , and geometric multiplicity as
Definition
Let be an endomorphism of a finite-dimensional vector space over , and let . The algebraic multiplicity of is the largest natural number such that divides in . This largest exponent exists because is a root, the factor theorem gives one factor , and the nonzero polynomial has finite degree and unique factorization.
The geometric multiplicity of is
The eigenspace is a nonzero subspace of the finite-dimensional space , so this dimension is defined and is at least .
Depends on
- The basis-independent characteristic polynomial $\chi_T$ of an endomorphism of a finite-dimensional space, including $\chi_T=1$ in dimension zero
- Eigenvalues, eigenvectors, eigenspaces $E_\lambda(T)=\ker(T-\lambda I)$, and the spectrum $\sigma_F(T)$ of an endomorphism
- Factor theorem over a commutative ring
- For every field $F$, $F[x]$ is a unique factorisation domain
- Finite-dimensional vector space, and its dimension $\dim_F V$; infinite-dimensional means having no finite basis
Used by
- An eigenvalue of algebraic multiplicity one has a one-dimensional eigenspace Corollary
- beginpmatrix2&10&2 endpmatrix has algebraic multiplicity two but geometric multiplicity one at 2 Example
- The scalar matrix 2I₂ has algebraic and geometric multiplicity two at 2 Example
- An endomorphism is diagonalisable if and only if its characteristic polynomial splits and every eigenvalue's geometric multiplicity equals its 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
- 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
- The geometric multiplicity of an eigenvalue does not exceed its algebraic multiplicity Theorem
Dependency tree · two levels
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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)