Construct the similarity classes of matrices having a given dimension and minimal polynomial

Construct (representatives of) the similarity classes of 6 \times 6 matrices over \mathbb{Q} having minimal polynomial m(x) = (x+2)^2(x-1).


Recall that the minimal polynomial of a matrix is its divisibility-largest invariant factor (Prop. 13 in D&F), that the characteristic polynomial of a matrix is the product of its invariant factors (Prop. 20 in D&F) and that the degree of the characteristic polynomial of a matrix is the dimension of the matrix (By the definition of the characteristic polynomial). To construct the rational canonical forms of 6 \times 6 matrices having minimal polynomial m(x), it suffices to construct all the possible lists of invariant factors which terminate in m(x) and whose product has degree 6.

Note that m(x) has five nontrivial divisors over \mathbb{Q}: x+2, (x+2)^2, x-1, (x+2)(x-1), and (x+2)^2(x-1). These are the possible next-to-last entries in a list of invariant factors which terminates in m(x). Note that for each of these divisors, the remaining invariant factors are determined by degree considerations except in the case (x+2)(x-1), where we have two possible choices for the third invariant factor.

The lists of invariant factors are as follows.

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

The corresponding similarity class representatives (i.e. rational canonical forms) are as follows.

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