This is the definition I have for eigenspace:
Let $\lambda$ be an eigenvalue of $A \in Mn(\mathbb C)$. Then
{$v \in \mathbb C^n|Av= \lambda v$}$=W_\lambda$
is a subspace of $\mathbb C^n$, called the eigenspace associated to $\lambda$.
The proof shows that $(0,0,.....,0)$ is in $W_\lambda$. But isn't an eigenvector supposed to be nonzero? How then is $(0,0,....,0)$ in $W_\lambda$?
Or maybe this definition is incomplete, and an eigenspace contains the zero vector?
$\endgroup$1 Answer
$\begingroup$The eigenspace corresponding to $\lambda$ is by definition the solution space of $A-\lambda I$, and hence it always contains the zero vector. The eigenvector/s corresponding to $\lambda$ are the non-zero vectors from the corresponding eigenspace.
$\endgroup$ 5
