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Math Help - JNF of nilpotent matrices

  1. #1
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    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?
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  2. #2
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    Quote Originally Posted by Alex314 View Post
    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 x^4 , the question now is: what's its minimal polynomial m(x)?

    For example, if 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 \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

    \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
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  3. #3
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    Thanks alot, I can see how to do it now.

    Alex
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