find all possible Jordan Normal Forms of A

Nov 2009
717
133
Wahiawa, Hawaii
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

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

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
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
\(\displaystyle \begin{bmatrix}-1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 &0 & 2\end{bmatrix}\)

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

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