Answer to Question #155566 in Linear Algebra for Wahit panda

Question

Find all possible jordan canonical form for the matrices whose characteristics polynomial p(t) and minimal polynomial m(t) are p(t)=(t-2)⁴(t-3)² m(t)= (t-2)²(t-3)²

Answer 1

First of all we remark that there is 2 eigenvalues - $Sp(A)= \{2; 3 \}$. Secondly, as characteristic polynomial is of degree 6, the matrix size is $6 \times6$. Let's study each of these eigenvalues separately :

• Characteristic subspace of $\lambda=3$ is of dimension 2 (by looking at the degree of $(x-3)$ in the characteristic polynomial). Therefore there is only two possible Jordan forms associated to this case : $\begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix}$ or $\begin{pmatrix} \lambda & 1 \\ 0 & \lambda \end{pmatrix}$. But the first case is eliminated, as that woud mean that $A|_{V_3}$ ($A$ restricted to the characteristic subspace associated to $\lambda=3$) is diagonalizable. But as the minimal polynomial of $A$ contains a term $(x-\lambda)^2$, it can not be diagonalizable. Therefore the only possible Jordan form is $\begin{pmatrix} \lambda & 1 \\ 0 & \lambda \end{pmatrix} =\begin{pmatrix} 3 & 1 \\ 0 & 3 \end{pmatrix}$.
• Characteristic subspace of $\lambda=2$ is of dimension 4. Therefore the possible Jordan forms are : $\begin{pmatrix} \lambda & 0 &0&0 \\ 0 & \lambda &0&0 \\0&0&\lambda&0 \\ 0&0&0&\lambda\end{pmatrix}$ , $\begin{pmatrix} \lambda & 1 &0&0 \\ 0 & \lambda &0&0 \\0&0&\lambda&0 \\ 0&0&0&\lambda\end{pmatrix}$ ,$\begin{pmatrix} \lambda & 1 &0&0 \\ 0 & \lambda &0&0 \\0&0&\lambda&1 \\ 0&0&0&\lambda\end{pmatrix}$ , $\begin{pmatrix} \lambda & 0 &0&0 \\ 0 & \lambda &1&0 \\0&0&\lambda&1 \\ 0&0&0&\lambda\end{pmatrix}$ and $\begin{pmatrix} \lambda & 1 &0&0 \\ 0 & \lambda &1&0 \\0&0&\lambda&1 \\ 0&0&0&\lambda\end{pmatrix}$. The first, fourth and fifth forms are excluded by studying the minimal polynomial : first form is associated to a matrix that is diagonalizable when is restricted to $V_2$ , the fourth and fifth would have the minimal polynomial containing the terms $(x-\lambda)^3$ and $(x-\lambda)^4$ respectively. Thus, There is only two possible Jordan forms, the second and the third.

Therefore the possible Jordan forms are :

$\begin{pmatrix} 2 & 1 &0&0 &0&0\\ 0 & 2 &0&0&0&0 \\0&0&2&1&0&0 \\ 0&0&0&2&0&0\\0&0&0&0&3&1\\0&0&0&0&0&3\end{pmatrix}$ , $\begin{pmatrix} 2 & 1 &0&0 &0&0\\ 0 & 2 &0&0&0&0 \\0&0&2&0&0&0 \\ 0&0&0&2&0&0\\0&0&0&0&3&1\\0&0&0&0&0&3\end{pmatrix}$ + the possible permutation of Jordan blocks.