Eigenvalues and Eigenvectors

• $\lambda\in \lambda(A) \Leftrightarrow A-\lambda I \text{ is singular} \Leftrightarrow \text{det}(A-\lambda I)=0$
• $\{x\neq0|x\in \mathcal{N}(A-\lambda I)\}$ is the set of all eigenvectors associated with $\lambda$.  
• $\mathcal{N}(A-\lambda I)$ is the eigenspace for A.