The geometric multiplicity of an eigenvalue is the dimension of its eigenspace: the subspace of vectors v such that (A − λI)v = 0. It counts the maximum number of linearly independent eigenvectors associated with that eigenvalue.

Principle

Geometric multiplicity = dim ker(A − λI); it is at least 1 for an eigenvalue λ and never exceeds the algebraic multiplicity of λ.

Demonstration

For A = [[2,1,0],[0,2,1],[0,0,2]] the characteristic polynomial is (λ−2)^3 so the algebraic multiplicity of 2 is 3, but ker(A−2I) is 1-dimensional (all eigenvectors are scalar multiples of [1,0,0]^T), so the geometric multiplicity of 2 is 1.

Misapplication

Assuming geometric multiplicity equals algebraic multiplicity without verification; treating generalized eigenvectors as part of the eigenspace; or expecting eigenvectors from different eigenvalues to be linearly dependent in general.

Consequence

Geometric multiplicities determine how many independent eigenvectors are available; a matrix is diagonalizable iff for every eigenvalue its geometric multiplicity equals its algebraic multiplicity.

Reversal

Maximal geometric multiplicity for an eigenvalue occurs when its geometric multiplicity equals its algebraic multiplicity, the opposite extreme from a defective eigenvalue where geometric multiplicity is strictly smaller.

Boundary

Defined with respect to the base field and only for eigenvalues that lie in that field; over extensions the eigenspace and its dimension may change. In infinite-dimensional settings eigenspaces may be infinite-dimensional and require separate treatment.

Semantic Tension

Tension with algebraic multiplicity: algebraic is a root-counting invariant, geometric is a linear-algebraic dimension; both constrain but do not determine each other except by the inequality 1 ≤ geometric ≤ algebraic.

Synthesis

Geometric multiplicity measures the linear independence available from eigenvectors for a given eigenvalue; it controls the dimension of eigenspaces and, together with algebraic multiplicity, governs diagonalizability and Jordan structure.