Eigenspaces

The set of eigenvectors corresponding to eigenvalue $\lambda$ (and $\vec{0}$) is called the eigenspace of A corresponding to $\lambda$.

• We also include $\vec{0}$ in the set so that it’s a subspace.
• Denoted by $E_{\lambda }(A)$.

These vectors satisfy:

• $(A-\lambda I)\vec{x}=0$ (by definition).
• So this set of vectors (along with $\vec{0}$) is basically the null space of $A-\lambda I$.

$$E_{\lambda }(A)=\text{Null}(A-\lambda I)$$