Suppose $V$ is a finite dimensional vector space over a field , and suppose $L\colon V\to V$ is a linear map. Suppose also that is an eigenvalue of $L$ , that is, $\operatorname{det}( L - \lambda I)=0$ .

The algebraic multiplicity, denoted by $A_\lambda(L)$ , of $\lambda$ is the multiplicity of the root $\lambda$ to the polynomial $\operatorname{det}( L - \lambda I)=0$ . The geometric multiplicity of $\lambda$ , denoted by $G_\lambda(L)$ , is the dimension of $\ker ( L - \lambda I)$ , the eigenspace of $\lambda$ .