Suppose we have a real $n\times n$ matrix $A$ with spectrum $\sigma(A)=\{\lambda_1,\lambda_2\}$ (with $\lambda_1, \lambda_2$ discrete). Also, we have $alg\,mult\, \lambda_i\neq geo\, mult \,\lambda_i $ for $i=1,2$.
We consider the vector space $$V=\left\{\begin{bmatrix} a_1 & \cdots & a_n \end{bmatrix}^T:a_1, \ldots, a_n \in \mathcal F\right\},$$ where $\mathcal F$ is a field.
Also, we consider the eigenspace $$V_\lambda=\{v \in V: A\cdot v=\lambda \cdot v\}.$$ Since we can find generalized eigenvectors (associated with $\lambda_i$), we have for sure a generalized eigenspace , which is associated with the eigenvalue $\lambda_i$, let's say $E_{\lambda_i}$.
Questions
Is there any relationship between $V_{\lambda_i}$ and $ E_{\lambda_i}$?
Is there any relationship among $V, V_{\lambda_i},E_{\lambda_i}$ (maybe something with direct sum) ?
Can we analyze matrix $A$ in terms of projections $P_{\lambda_i}$ onto $V_{\lambda_i}$?
Do we need any extra assumptions for $\mathcal F$ in order to answer all the questions above?
I know these are fundamental concepts, but I am trying to figure them out, especially when eigenvalues are not discrete and their algebraic multiplicity does not equal its geometric one.
Thank you for your time.
P.S. Feel free to edit my question if any of the statements are mistaken or do not make sense.
Yes. $V_{\lambda_i} \subseteq E_{\lambda_i}$, with equality iff the algebraic and geometric multiplicities of $\lambda_i$ are equal
If $\mathcal F$ is algebraically closed, then we necessarily have $$ V = \bigoplus_{i=1}^k E_{\lambda_i} $$
Potentially. Not generally with orthogonal projections, though, so this analysis wouldn't be as useful as it is for normal or self-adjoint matrices
I think that should cover it. Feel free to comment if you'd like any clarification on some of these points.