Number of Jordan Canonical Forms Associted With the the Characteristic Polynomial of a Linear Transformation

Let

\[T\]

be a linear operator with characteristic polynomial

\[f(\lambda)=(a_1- \lambda_1)^{n_1} (a_2- \lambda_2)^{n_2}... (a_k- \lambda_k)^{n_k} \]

and minimum polynomial

\[p(\lambda)=(a_1- \lambda_1)^{m_1} (a_2- \lambda_2)^{m_2}... (a_k- \lambda_k)^{m_k} \]

with

\[m_i \leq n_i, \: i=1,2,...,k\]

With this characteristic polynomial we can find all possible Jord an canonical matrices.
The Jordan Form matrix for

\[f(\lambda)=(2- \lambda)^2(5-\lambda)^2\]

\[ \left| \begin{array}{ccccc} 2 & 1 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 2 & 1 & 0 \\ 0 & 0 & 0 & 5 & 1 \\ 0 & 0 & 0 & 0 & 5 \end{array} \right| \]

.
This matrix consists of 'Jordan blocks' along the main diagonal, a 3 x 3 matrix in the upper left, and a 2 x 2 matrix in the lower right.
These submatrices can be further subdivided, with the proviso that the largest submatrix, or equal largest, is in the upper left left of the submatrix. Hence we can subdivide the 3 x 3 matrix into a 3 matrix, a 2 x 1 matrix or a 1 x 1 x 1 matrix (but not a 1 x 2 matrix).
The 2 x 2 matrix in the lower left can only be subdivided into a 3 matrix or a 1 x 1 matrix.
He can write the whole matrix as
(3) x (2)
(2 x 1) x (2)
(3) x (1 x 1)
(2 x 1) x (1 x 1)
(1 x 1 x 1) x (1 x 1)