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

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