9. Eigenspaces

Definition. For a square matrix $A$ and scalar $\lambda$, the eigenspace of $\lambda$ consists of all v with $\tau _ A ($v$) = \lambda$   v. We can call the eigenspace $E _ \lambda$. In other words, $E _ \lambda$ consists of all eigenvectors for $\lambda$ and also the zero vector.

One advantage of talking about eigenspaces instead of eigenvectors is that it's OK to talk about $E _ \lambda$ even when $\lambda$ is not an eigenvalue of $A$; in that case, $E _ \lambda = \{$0$\}$.

Another advantage is that it's easier to think about examples such as $A=I$, where $E _ 1$ is the whole space, or $A = \matp{rrr}{2&0&0\\ 0&2&0\\ 0&0&3}$, where $E _ 2$ is a plane.

Problem U-14. Explain: $E _ \lambda$ is the nullspace of $A - \lambda I$, or equivalently, the kernel of $\tau _ {A - \lambda I}$. Therefore $E _ \lambda$ is always a subspace.

Problem U-15. In each case, make up an example of a $3 \times 3$ matrix so that $E _ 5$ is (a) $\{$0$\}$, (b) a line, (c) a plane, (d) all of $\mathbb{R}^ 3$. (Suggestion: Use diagonal matrices.)