Definition: The algebraic multiplicity of an eigenvalue $\lambda$ is its multiplicity as a root of the characteristic polynomial. The geometric multiplicity is the dimension of the eigenspace $E_\lambda$. Always: geometric multiplicity $\leq$ algebraic multiplicity.
**Definition:** The **algebraic multiplicity** of an eigenvalue $\lambda$ is its multiplicity as a root of the characteristic polynomial. The **geometric multiplicity** is the dimension of the eigenspace $E_\lambda$. Always: geometric multiplicity $\leq$ algebraic multiplicity.