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?