# multiplicity of eigenvalue

Suppose $V$ is a finite dimensional vector space over a field $\mathbbmss{F}$, and suppose $L\colon V\to V$ is a linear map. Suppose also that $\lambda\in\mathbbmss{F}$ 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 $\lambda$.

Title multiplicity of eigenvalue MultiplicityOfEigenvalue 2013-03-22 15:15:15 2013-03-22 15:15:15 matte (1858) matte (1858) 5 matte (1858) Definition msc 15A18 geometric multiplicity algebraic multiplicity