Question: I want to determine the JCF for a matrix with characteristic polynomial $(x-3)^5$ and minimal polynomial $(x-3)^3$. So, I know we have a $5\times 5$ matrix by the degree of char poly, and I know that we can't have any Jordan Blocks of size $4$ or $5$ by the degree of the minimal polynomial. My question is the since the degree of the min poly is $3$, do we HAVE to have a Jordan Block of size $3$? If so, then we get $2$ possibilities: first, a Jordan Block of size $3$ and two Jordan Blocks of size $1$, and second a Jordan block of size $3$ and a Jordan Block of size $2$.
So, if I was given some $5\times 5$ matrix satisfying the characteristic poly and minimal poly above, I could just find the dimension of the null space of $3I-A$ (where $A$ is the given matrix), and if that dimension is $3$, then it must be the first possibility above for the JCF and if the dimension of the null space of $3I-A$ is $2$, then the JCF must be the second possibility from above… right?
I am trying to get my head around JCF stuff, and the two books I have on linear algebra don't touch on it. So, if anyone could let me know if I am right, or if I said anything incorrect, that would be great. Moreover, if you think there is anything else that could be added relating the minimal poly, char poly, JCF, Eigenvectors, Eigenvalues, etc., please feel free to share! 🙂
Thank you much!
Best Answer
For such an information the companion matrix is $$\left[ \begin{array}{ccccc} 0&0&27&0&0\\ 1&0&-27&0&0\\ 0&1&9&0&0\\ 0&0&0&3&0\\ 0&0&0&0&3 \end{array} \right],$$ which has canonical form $$\left[ \begin{array}{ccccc} 3&1&0&0&0\\ 0&3&1&0&0\\ 0&0&3&0&0\\ 0&0&0&3&0\\ 0&0&0&0&3 \end{array} \right].$$