# Possibilities for the Jordan normal form

I need to show:
How many possibilities for the Jordan normal form (up to order of the Jordan blocks on the diagonals) there are for matrices $B \in M_5 (\mathbb C)$ with $x_B(\lambda)=-(\lambda-2)^2(\lambda-3)^3$

The Eigenvalues are $2$ and $3$.
The Eigenvalue $2$ has the algebraic multiplicity 2 and the geometric multiplicity $2$ or $1$.
The Eigenvalue $3$ has the algebraic multiplicity 3 and the geometric multiplicity $3$ or 2 or $1$.

So I tried showing all the possibilites with:

1) $diag (J_1 (2), J_1 (2), J_1(3), J_1(3), J_1(3))$

2) $diag (J_2 (2), J_1(3), J_1(3), J_1(3))$

3) $diag (J_2 (2), J_3(3))$

4) $diag (J_1 (2), J_1 (2), J_3(3))$

5)$diag (J_2 (2), J_2(3), J_1(3))$

6) $diag (J_1 (2), J_1 (2), J_2(3), J_1(3))$

I am not sure if this is right though..

• I'm assuming you mean $\chi_B(\lambda)=-(\lambda-2)^2(\lambda-3)^3$? – Brian Fitzpatrick Jul 11 '17 at 19:51
• Correct... Sry my mistake – PhysX Jul 11 '17 at 19:52

Yes, this is correct. Let $\DeclareMathOperator{am}{am}\am_B(\lambda)$ and $\DeclareMathOperator{gm}{gm}\gm_B(\lambda)$ denote the algebraic and geometric multiplicities of an eigenvalue $\lambda$ of $B$ respectively. Then the table of eigenvalues of $B$ along with their algebraic multiplicities and possible geometric multiplicities is $$\begin{array}{c|c|c} \lambda & \am_B(\lambda) & \gm_B(\lambda) \\ \hline 2 & 2 & 1,2 \\ 3 & 3 & 1,2,3 \end{array}$$ There are thus six possible combinations of geometric multiplicities of the eigenvalues of $B$. These six possible combinations yield exactly the six Jordan forms you have listed.
Note, however, that in general a choice of geometric multiplicities can yield more than one possible Jordan form. For example, both \begin{align*} J_2(\lambda)\oplus J_2(\lambda) && J_3(\lambda)\oplus J_1(\lambda) \end{align*} have characteristic polynomial $\chi(t)=(t-\lambda)^4$ and hence have exactly one eigenvalue with geometric multiplicity two.