Compute all the matrices having a given characteristic polynomial

Find all the possible rational canonical forms of matrices having characteristic polynomial c(x) = x^2(x^2+1)^2.


Let F be a field. Our solution depends on how the polynomial x^2+1 factors over F.

Suppose x^2+1 is irreducible in F. (For instance, in \mathbb{Q} or \mathbb{Z}/(3).) If A is a matrix with characteristic polynomial c(x), then the minimal polynomial of A must divide c(x) and be divisible by x and by x^2+1. There are four such divisors, and with the minimal polynomial chosen, in each case the remaining invariant factors are determined. These are as follows.

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

The corresponding rational canonical forms are as follows.

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

Every 6 \times 6 matrix over F whose characteristic polynomial is c(x) is similar to exactly one matrix on this list.

Now suppose x^2+1 factors over \varepsilon in F. (For instance, in \mathbb{C} or \mathbb{Z}/(5).) Since x^2+1 has degree 2, there is only one way this can happen- namely, F contains both roots of x^2+1. If these roots are called \varepsilon and \eta, then comparing coefficients in (x-\eta)(x-\varepsilon) = x^2+1, we see that -\eta = \varepsilon. Now the irreducible factorization of c(x) over F is c(x) = x^2(x-\eta)^2(x+\eta)^2. Now if A is a matrix with characteristic polynomial c(x), then the minimal polynomial of A must divide c(x) and is divisible by x(x-\eta)(x+\eta). There are eight such divisors of c(x), and for each choice of the minimal polynomial, the remaining invariant factors are determined. The possible lists of invariant factors for A are as follows.

  1. x(x-\eta)(x+\eta), x(x-\eta)(x+\eta)
  2. (x-\eta)(x+\eta), x^2(x-\eta)(x+\eta)
  3. x(x+\eta), x(x-\eta)^2(x+\eta)
  4. x(x-\eta), x(x-\eta)(x+\eta)^2
  5. x, x(x-\eta)^2(x+\eta)^2
  6. x-\eta, x^2(x-\eta)(x+\eta)^2
  7. x+\eta, x^2(x-\eta)^2(x+\eta)
  8. x^2(x-\eta)^2(x+\eta)^2

The corresponding rational canonical form matrices are as follows.

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

Every 6 \times 6 matrix over F having characteristic polynomial c(x) is similar to exactly one matrix in this list.

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