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

1. 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

2. Originally Posted by matty888
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?