geometric multiplicity

Algebraic multiplicity counts how often an eigenvalue appears as a root of a polynomial; geometric multiplicity counts something more concrete — how many genuinely independent directions get scaled by that eigenvalue. It is the number of linearly independent eigenvectors you can find for lambda, which is the same as the dimension of the eigenspace.

Formally, the geometric multiplicity of lambda is dim ker(A - lambda I), the dimension of the null space of A - lambda I. By the rank-nullity theorem this equals n - rank(A - lambda I). A fundamental theorem says geometric multiplicity is always at least 1 (an eigenvalue has at least one eigenvector) and never exceeds the algebraic multiplicity: 1 <= geom <= alg.

The gap between the two is the whole drama of diagonalizability. When geom = alg for every eigenvalue, you have enough eigenvectors to fill out a basis and the matrix diagonalizes. When geom < alg for even one eigenvalue, the matrix is defective: it is short of eigenvectors and cannot be diagonalized — you must climb to generalized eigenvectors and the Jordan form to complete the picture.