Two questions:
How does the nilpotent index $k$ of a linear transformation L on a vector space of dimension $n$ relate to possible Jordan Canonical Forms?
My understanding is that a Jordan block can have a size of at most the nilpotent index $k$ so for example a vector space of dimension 5 and a linear transformation of nilpotent index 3 means that the possible ordered Jordan Canonical Forms are 5 in total 3-2, 3-1-1, 2-2-1, 2-1-1-1, 1-1-1-1-1. Is this correct?
How does the dimension of an eigenspace of matrix A representing a linear transformation L on a vector space of dimension $n$ relate to the possible Jordan Canonical Forms?
My understanding is that the dimension of the eigenspace is equal to the number of Jordan Blocks so for an eigenspace of dimension 3 and a vector space of dimension 5, the number of Jordan Blocks will be 3 and the possible ordered forms will be either 3-1-1 or 2-2-1. Is this correct?
If by nilpotent index you mean the smallest $k$ such that $L^k = 0$, then yes both of those statements are correct and are the only constraints implied by that information.