# All Possible Jordan Canonical Forms Given Characteristic Polynomial

I am given the characteristic polynomial $x^2(x^2-1)$ and am asked to find all possible jordan canonical forms. What I have so far is:

Possible elementary divisors are: 1) $x,x,(x+1),(x-1)$, 2) $x,x,(x+1)(x-1)$, 3) $x^2,(x+1)(x-1)$, and 4) $x^2(x+1)(x-1)$. I therefore got the possible Jordan forms as:

1)=2) \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{matrix}

and

3)=4) \begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{matrix}

I'm really unsure if these are correct, so any insight would be greatly appreciated!

• The use of the term elementary divisors suggests you're looking for a way to relate JNF with smith normal form. Is this the case? Mar 12, 2014 at 15:04

They are correct. What the characteristic polynomial $$x^2(x-1)(x+1)$$tells you is that there are at least $3$ cages, of which the cages for eigenvalues $1$ and $-1$ have size $1$. This only leaves the eigenvalue $0$, for which you have $2$ options:
1. There are $2$ linearly independent eigenvectors for the eigenvalue $0$. In that case, your matrix has $4$ eigenvectors and can be diagonalized.
2. There is only $1$ linearly independent eigenvector for $0$. This means that the cage for $0$ is $2\times 2$ and the Jordan form is as your second matrix suggests.
• While basically correct, you should avoid saying "there are $2$ eigenvectors...". There are always infinitely many eigenvectors. Mar 12, 2014 at 15:10
• How about "$2$ linearly independent eigenvectors"? Mar 12, 2014 at 15:13