The algebraic multiplicity of an eigenvalue λ of a linear operator on a finite-dimensional vector space is its multiplicity as a root of the characteristic polynomial det(A − λI).

Principle

It counts how many times an eigenvalue appears algebraically in the characteristic equation and constrains the sizes of Jordan blocks associated with that eigenvalue.

Demonstration

For A = [[λ,1],[0,λ]] the characteristic polynomial is (λ − λ)^2 so λ has algebraic multiplicity 2; the matrix has a single Jordan block of size 2, showing algebraic multiplicity exceeds geometric multiplicity (which is 1).

Misapplication

Assuming algebraic multiplicity equals the number of linearly independent eigenvectors (geometric multiplicity) in non-diagonalizable cases; or applying the notion unchanged to operators on infinite-dimensional spaces without spectral theory qualifiers.

Consequence

Knowing algebraic multiplicities determines the total size of generalized eigenspaces and, together with geometric multiplicities, fixes the Jordan normal form and the degree of the minimal polynomial.

Reversal

Geometric multiplicity is the dimension of the eigenspace ker(A − λI); it is always ≤ algebraic multiplicity and equals it exactly when the operator is diagonalizable at λ.

Boundary

Definition requires a characteristic polynomial: it is intrinsic for finite-dimensional linear operators over fields where the polynomial factors (commonly algebraically closed fields); for continuous spectra or operators on infinite-dimensional spaces the concept must be replaced by spectral multiplicity notions.

Semantic Tension

Algebraic multiplicity versus spectral/measure-theoretic multiplicity: algebraic multiplicity is a discrete root count in finite dimensions, while spectral multiplicity in functional analysis describes invariant subspace multiplicities of continuous spectrum components.

Synthesis

Algebraic multiplicity measures how many algebraic factors (λ − λ0) appear in the characteristic polynomial, determining the total count of generalized eigenvectors and constraining Jordan block structure, distinct from but related to geometric multiplicity.