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

Construct (representatives of) the similarity classes of 6 \times 6 matrices over \mathbb{C} having characteristic polynomial c(x) = (x^4-1)(x^2-1).


First, note that c(x) factors over \mathbb{C} as c(x) = (x+1)^2(x-1)^2(x+i)(x-i).

Recall that the characteristic polynomial of a matrix is the product of its invariant factors, that the minimal polynomial is the divisibility-largest invariant factor, and that the characteristic polynomial divides a power of the minimal polynomial (that is, since \mathbb{C}[x] is a UFD, each factor of the characteristic polynomial appears in the factorization of the minimal polynomial).

If A is a matrix having characteristic polynomial c(x), then the minimal polynomial of A is one of the following.

  1. (x+i)(x-i)(x+1)(x-1)
  2. (x+i)(x-i)(x+1)^2(x-1)
  3. (x+i)(x-i)(x+1)(x-1)^2
  4. (x+i)(x-i)(x+1)^2(x-1)^2

In each case, the remaining invariant factors are determined. The possible lists of invariant factors are thus as follows.

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

The corresponding matrices are thus as follows.

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

Every 6 \times 6 matrix over \mathbb{C} with characteristic polynomial c(x) is similar to exactly one matrix in this list.

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