Number of Jordan canonical form of a matrix

Question

Let, $A\in M(3,C)$. Assume that the characteristic & minimal polynomial of $A$ are known. Then what is the number of possible Jordan form of $A$ and how?

What changes if we replace $C$ by $R$ or if we replace 3 by 4 ?

Answer 1

Hint: whether it's 3 or 4 will affect your answer, the choice of R vs. C will not.

Note that the degree of a factor in the characteristic polynomial is the multiplicity of the corresponding eigenvalue. That is, the degree is the sum of the sizes of the blocks associated with that eigenvalue.

The degree of a factor in the minimal polynomial is the size of the largest associated Jordan block.

An example for the $4 \times 4$ case: $$ \pmatrix { 1&1\\ &1\\ &&1&1\\ &&&1 }, \quad \pmatrix { 1&1\\ &1\\ &&1\\ &&&1 } $$

Answer 2

Hint: $$ { 3 \choose 1}+{ 3 \choose 2}+{ 3 \choose 3} $$