# JNF of nilpotent matrices

• Mar 31st 2010, 02:04 PM
Alex314
JNF of nilpotent matrices
A question on one of my past exam papers is

1. List all possible Jordan normal forms of nilpotent 4 x 4 matrices

How would you go about doing this?
• Mar 31st 2010, 06:23 PM
tonio
Quote:

Originally Posted by Alex314
A question on one of my past exam papers is

1. List all possible Jordan normal forms of nilpotent 4 x 4 matrices

How would you go about doing this?

The char. polynomial of a nilpotent 4x4 matrix is $\displaystyle x^4$ , the question now is: what's its minimal polynomial$\displaystyle m(x)$?

For example, if $\displaystyle m(x)=x^2\Longrightarrow$ there's at least one Jordan Block (JB) of order 2, so you have two possibilities for the JNF of a matrix like this:

either $\displaystyle \begin{pmatrix}0&1&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\ end{pmatrix}$ --- one 2x2 JB + two 1x1 JB's , or

$\displaystyle \begin{pmatrix}0&1&0&0\\0&0&0&0\\0&0&0&1\\0&0&0&0\ end{pmatrix}$ --- two 2x2 JB's.

Well, now check the other possibilities for the minimal polynomial.

Tonio
• Apr 1st 2010, 02:49 AM
Alex314
Thanks alot, I can see how to do it now.

Alex