# Thread: find all possible Jordan Normal Forms of A

1. ## find all possible Jordan Normal Forms of A

This might be in the wrong forum
But it is from a DE class

Suppose that A is a matrix whose characteristic polynomial is
$$(\lambda-2)^2(\lambda+1)^2$$
find all possible Jordan Normal Forms of A (up to permutation of the Jordan blocks).

ok i have been looking at examples so pretty fuzzy on this
for the roots are 2 and -1

so my first stab at this is

$\left[\begin{array}{c} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 &2 \end{array}\right]$

2. ## Re: find all possible Jordan Normal Forms of A

Yes, that is one possible Jordan form. The general Jordan form has blocks for distinct eigenvalues with either 1 or 0 above each eigenvalue in each block. The other possible Jordan forms are
$\displaystyle \begin{bmatrix}-1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 &0 & 2\end{bmatrix}$

$\displaystyle \begin{bmatrix}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2\end{bmatrix}$

$\displaystyle \begin{bmatrix}-1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2\end{bmatrix}$