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Math Help - Matrix Qn!!Plz help!!Very Stuck

  1. #1
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    Matrix Qn!!Plz help!!Very Stuck

    Find a Jordan basis for the matrix


    C =

    and write down the corresponding Jordan form.
    Plz help!!Very Stuck
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  2. #2
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    Quote Originally Posted by matty888 View Post
    Find a Jordan basis for the matrix


    C =

    and write down the corresponding Jordan form.
    Plz help!!Very Stuck
    Obviously, the eigenvalues of C are 3,3,3,3. Let I_4 be the identity matrix and consider A=C-3I_4, by direct computation, you see that rank(A)=2,rank(A^2)=1,rank(A^3)=0. These data say the following:
    the number of Jordan blocks of at least size 1= 4-rank(A)=4-2=2
    the number of Jordan blocks of at least size 2= rank(A)-rank(A^2)=2-1=1
    the number of Jordan blocks of at least size 3= rank(A^2)-rank(A^3)=1-0=1
    the number of Jordan blocks of at least size 4= rank(A^3)-rank(A^4)=0
    So the Jordan form of A is J=\left [\begin{array}{cccc}0&1&0&0\\ 0&0&1&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right ] and hence the Jordan form of C is J+3I_4.

    I don't understand the definition of "Jordan basis", could you tell me?
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