(Visual) intuition behind the geometric multiplicity of eigenvalues

Question

I computed the eigenvalues- and spaces for the following 3x3 matrix: $$ \begin{pmatrix} 4&2&3 \\ -1 & 1 & -3\\ 2 & 4 & 9\\ \end{pmatrix} $$ The eigenvalue 3 has an algebraic multiplicity of 2, so its geometric multiplicity can be 1 or 2. In fact, its geometric multiplicity is indeed 2 as the eigenspace is spanned by the vectors $\begin{pmatrix}-2 \\ 1 \\ 0\end{pmatrix}$ , $\begin{pmatrix}-3 \\ 0 \\ 1\end{pmatrix}$.
Calculating all this stuff is easy for me, I just can't wrap my head around what I'm supposed to do with the information I gain from the geometric multiplicity. What's the difference between a geometric multiplicity of 1 and 2 in $\mathbb{R}^3$? Does the geometric multiplicity of 2 in my case just mean that all eigenvectors associated with my eigenvalue of 3 lie in a plane instead of on a line?
Thanks in advance!

Answer 1

Asserting that the geometric multiplicity of the eigenvalue $3$ is $2$ means that the dimension of the eigenspace corresponding to the eigenvalue $3$ is $2$. So, that eigenspace is a plane passing through the origin. If the geometric multiplicity was $1$, then the eigenspace would be a line passing through the origin.