This has been a burning problem in my head for a while now. Any help/suggestions are greatly appreciated. I'll use a concrete 3x3 matrix as an example, but I'd like to know thoughts especially for 4x4 and, if possible NxN matrices as well. The question originally popped into my head while I was learning about the Segre Classification of spacetimes.
Given a 3x3 matrix $A$
$A=\begin{bmatrix} a1 & a2 & a3 &\\a4 & a5 & a6 \\a7 & a8 & a9\end{bmatrix}$.
or possibly a symmetric matrix $B$ (whichever things can be said about. I'd like to know both cases if possible)
$B=\begin{bmatrix} b1 & b2 & b3 &\\b2 & b4 & b5 \\b3 & b5 & b6\end{bmatrix}$.
The question is: Is it possible to determine the dimension of the eigenspace from some relation of the a's or b's? (Various hopeful clarifications are given below)
For example, I know that I can determine repeated roots (repeated eigenvalues) from just looking at the discriminant of the characteristic polynomial. That's one thing done in the Segre classification. I'd like to extend this idea and know if there are any invariants which characterize the sizes of Jordan blocks/dimensions of the eigenspaces which I can just calculate from $A$ or $B$ or powers of each. The trouble I have been having is the minimal polynomial which would normally determine some of this depends on the eigenvalues which change as a1...a9 etc are varied. The easiest example here that I can think of is:
$\tilde{A}=\begin{bmatrix} 2 & 1 & 0 &\\0 & 2 & 1 \\0 & 0 & 2\end{bmatrix}$.
Where the minimal polynomial is $(\lambda-2)^3$. But if I perturb a1->a1+epsilon, this instantly breaks and the minimal polynomial changes. Nevertheless, I believe there should be something in terms of the a's or b's to calculate out when exactly this happens.
Thanks