Find the possible Jordan canonical forms of a matrix having a given characteristic polynomial

Find all possible Jordan canonical forms of matrices having characteristic polynomial p(x) = (x-2)^3(x-3)^2.


The characteristic polynomial of a matrix A is the product of its elementary divisors. Distinct lists of elementary divisors correspond precisely to a choice, for each linear factor x-\alpha of p(x), of a partition of the exponent of x-\alpha in the factorization of p(x). The partitions of 3 are 3, 2+1, and 1+1+1, and the partitions of 2 are 2 and 1+1. The possible lists of elementary divisors for A are thus as follows.

  1. (x-2)^3, (x-3)^2
  2. (x-2)^2, x-2, (x-3)^2
  3. x-2, x-2, x-2, (x-3)^2
  4. (x-2)^3, x-3, x-3
  5. (x-2)^2, x-2, x-3, x-3
  6. x-2, x-2, x-2, x-3, x-3

The possible Jordan canonical forms are thus as follows.

  1. \begin{bmatrix} 2 & 1 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 0 & 3 \end{bmatrix}
  2. \begin{bmatrix} 2 & 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 0 & 3 \end{bmatrix}
  3. \begin{bmatrix} 2 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 0 & 3 \end{bmatrix}
  4. \begin{bmatrix} 2 & 1 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 3 \end{bmatrix}
  5. \begin{bmatrix} 2 & 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 3 \end{bmatrix}
  6. \begin{bmatrix} 2 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 3 \end{bmatrix}
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