The exponent of an eigenvalue as a root of the characteristic polynomial of a linear operator or matrix; equivalently, the power of the corresponding linear factor in the polynomial factorization.

Principle

Algebraic multiplicity is determined by the factorization of the characteristic polynomial: if (λ − λ0)^m divides the polynomial, then λ0 has algebraic multiplicity m.

Demonstration

For the 3×3 matrix with characteristic polynomial (λ−2)^2(λ−3), the eigenvalue 2 has algebraic multiplicity 2 and 3 has algebraic multiplicity 1; the characteristic polynomial records these counts even if eigenvectors are fewer.

Misapplication

Treating algebraic multiplicity as the count of independent eigenvectors leads to error when a repeated root corresponds to a defective eigenspace; conflating it with geometric multiplicity is a common misuse.

Consequence

Correct use yields the exponent counts needed for the Jordan canonical form and ensures the sum of algebraic multiplicities equals the degree of the characteristic polynomial.

Reversal

Reversing the concept yields geometric multiplicity: the dimension of the eigenspace rather than the root multiplicity in the polynomial.

Boundary

Applies to finite-dimensional linear operators or matrices over a field where a characteristic polynomial is defined; not directly meaningful for general infinite-dimensional operators or spectral notions without a polynomial.

Semantic Tension

Tension exists between algebraic multiplicity (a polynomial-root count) and geometric multiplicity (a vector-space dimension); they agree for diagonalizable operators but can differ otherwise.

Synthesis

Algebraic multiplicity is the polynomial algebra measure of how strongly an eigenvalue appears, which combined with eigenspace information determines canonical decompositions like Jordan form.