Eigenspaces: Intuition behind geometric multiplicity $\leq$ algebraic multiplicity? 
Theorem 6.6: Let $A$ be a square matrix, let $\gamma$ be an eigenvalue of $A$ with multiplicity $m$. Then the dimension of the eigenspace of $A$ corresponding to the eigenvalue $\gamma$ is less than or equal to $m$.

Could someone explain the intuition behind this theorem? I am not looking for a proof, one already exists on this site. Please give me intuition behind why its true.
Related: Eigenspace and polynomials?
Thanks.
 A: Here is how I think about this: consider the case of a nilpotent operator $A$ on a vector space $V$ of dimension $n$, so that $A^n=0$. The characteristic polynomial is $T^n$. The $0$-eigenspace of $A$ is just its kernel; thus, in this case, the theorem says that the dimension of the kernel of $A$ is not more than the dimension of $V$, which is obvious. In fact, the essential argument is already contained in this simple case.
You can view it like this: the evil operator $A$ is trying to kill the innocent citizens of $V$, but they're not going without putting up a fight. The unlucky citizens die at time $t=1$: those are the ones in the kernel of $A$. The luckiest citizens might survive up to time $t=n$. The theorem says that the set of unlucky citizens is contained in the set of all citizens.
A: You must be talking about the multiplicity of the eigenvalue as root of the characteristic polynomial (which is just one possible tool to find eigenvalues; nothing in the definition of eigenvalues says that this is the most natural notion of multiplicity for eigenvalues, though people do tend to assume that).
Now suppose you have an eigenspace for $\gamma$ of dimension $d$. Choosing a basis for the vector space that starts with a basis for this eigenspace, and is extended arbitrarily, you see that $A$ transformed to this basis is block upper triangular, starting with a diagonal block of size $d$ that is $\gamma$ times the ($d\times d$) identity. This block contributes a factor $(X-\gamma)^d$ to the characteristic polynomial$~\chi_A$, whence the multiplicity $m$ of $\gamma$ as root of $\chi_A$ is at least$~d$.
I admit this is proof, while you were not asking for one, but I find it intuitive enough. It also turns out to be the same proof in the question you linked to; what is wrong with the intuition from that answer?
