What is the number of non-conjugate $6 \times 6$ complex matrices having the characteristic polynomial $(x-5)^6=0$?

Question

Let $D$ be the $6 \times 6$ diagonal matrix with diagonal entries $5$.

Then all the $6 \times 6$ complex matrices which are diagonalizable to $D$ are conjugate to $D$ and hence to each other.

So I should find those matrices which aren't diagonalizable to $D$ but have same characteristic equation.

I think of those matrices whose all diagonal elements are $5$ but Geometric multiplicity $\neq$ Algebraic multiplicity for $5$. But still not getting any concrete idea.

What is the general way to approach?

Answer 1

These are distinguished by the Jordan forms, each of which consists of blocks of sizes which add up to $6$. So the number of possible Jordan forms is the sixth partition number $p(6)$.