I'm confused about how to find the possible dimension of an eigenspace given that a matrix has exactly one eigenvalue.
Suppose $A$ is a $3\times 3$ matrix, with exactly one eigenvalue $\lambda$. I assume I'm working over $\mathbb{R}$, so $\lambda$ is the only real root of the characteristic polynomial of $A$. Since this is a cubic, $\lambda$ has multiplicity $1$ or $3$.
Since the dimension of the eigenspace is at most the algebraic multiplicity of the eigenvalue, I think the dimension is either $0$ or $1$, or $0,1,2$ or $3$.
But the possible answers (it is a multiple choice question) are
- $1$
- $2$
- $3$
- $1$ or $2$
- $1$, $2$, or $3$
How can I more precisely determine the dimension?