algebraic multiplicity

When you factor the characteristic polynomial, each eigenvalue shows up as a root, and sometimes it shows up more than once. The algebraic multiplicity of an eigenvalue lambda is simply how many times the factor (x - lambda) appears in that factorization. If the characteristic polynomial is (x - 3)^2 (x - 5), then 3 has algebraic multiplicity 2 and 5 has algebraic multiplicity 1.

Precisely: for an n-by-n matrix A, write the characteristic polynomial p(x) = det(xI - A) as a product of linear factors over the algebraic closure. The algebraic multiplicity of lambda is the largest k such that (x - lambda)^k divides p(x). Because p has degree n, the algebraic multiplicities of all eigenvalues add up to exactly n (counting over C, where the polynomial always splits).

This number is the eigenvalue counted -with multiplicity-, and it sets a ceiling for the geometric side of the story. You can never have more independent eigenvectors for lambda than its algebraic multiplicity allows. When the two multiplicities match for every eigenvalue, the matrix is diagonalizable; when algebraic exceeds geometric, the matrix is defective and needs Jordan blocks to be tamed.