# (Visual) intuition behind the geometric multiplicity of eigenvalues

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?
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.
• Not sure this is correct. Algebraic multiplicity has obvious meaning - it refers to multiplicity of the root. Now - 3 is eigenvalue with algebraic multiplicity 2. But we do not know if there are 2 (independent) eigenvectors corresponding to this value. We know there is one. The other one can be (must be in fact) in the eigenspace corresponding to 3 i.e. $(A-3I)^2 x_2 = 0$ but $(A-3I)x_2 != 0$ Commented Jul 8, 2022 at 18:05