Find the possible Jordan canonical forms of matrices of dimension 2, 3, or 4 over CC

Find all the possible Jordan canonical forms of matrices of dimension 2, 3, or 4 over \mathbb{C}.


We begin by finding the possible lists of invariant factors, starting with the possible minimal polynomials. Recall that every polynomial of degree at least 1 over \mathbb{C} has a root in \mathbb{C}, so that every polynomial is a product of linear factors.

If A has dimension 2, then the characteristic polynomial of A has degree 2 and thus the minimal polynomial has degree at most 2. The possible minimal polynomials are thus x-\alpha, (x-\alpha)^2, and (x-\alpha)(x-\beta). In this case, with the minimal polynomial chosen the remaining invariant factors are determined. So the possible lists of invariant factors are as follows.

  1. x-\alpha, x-\alpha
  2. (x-\alpha)^2
  3. (x-\alpha)(x-\beta)

The corresponding lists of elementary divisors are as follows.

  1. x-\alpha, x-\alpha
  2. (x-\alpha)^2
  3. x-\alpha, x-\beta

And so the possible Jordan canonical forms are as follows.

  1. \begin{bmatrix} \alpha & 0 \\ 0 & \alpha \end{bmatrix}
  2. \begin{bmatrix} \alpha & 1 \\ 0 & \alpha \end{bmatrix}
  3. \begin{bmatrix} \alpha & 0 \\ 0 & \beta \end{bmatrix}

Again, if A has dimension 3, we can construct all the possible minimal polynomials, and in each case the remaining invariant factors are determined (in one case, without loss of generality). The possible lists of invariant factors are as follows.

  1. x-\alpha, x-\alpha, x-\alpha
  2. (x-\alpha)^2, x-\alpha
  3. (x-\alpha)^3
  4. (x-\alpha)(x-\beta), x-\alpha
  5. (x-\alpha)^2(x-\beta)
  6. (x-\alpha)(x-\beta)(x-\gamma)

The possible lists of elementary divisors are as follows.

  1. x-\alpha, x-\alpha, x-\alpha
  2. (x-\alpha)^2, x-\alpha
  3. (x-\alpha)^3
  4. x-\alpha, x-\beta, x-\alpha
  5. (x-\alpha)^2, x-\beta
  6. x-\alpha, x-\beta, x-\gamma

The possible Jordan canonical forms are then as follows.

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

If A has degree 4, evidently there are 11 possible minimal polynomials and 14 possible lists of invariant factors (in some cases, without loss of generality due to symmetry). The possible lists of invariant factors are as follows.

  1. x-\alpha, x-\alpha, x-\alpha, x-\alpha
  2. (x-\alpha)^2, (x-\alpha)^2
  3. (x-\alpha)^2, x-\alpha, x-\alpha
  4. (x-\alpha)(x-\beta), (x-\alpha)(x-\beta)
  5. (x-\alpha)(x-\beta), x-\alpha, x-\alpha
  6. (x-\alpha)^3, x-\alpha
  7. (x-\alpha)^2(x-\beta), x-\alpha
  8. (x-\alpha)^2(x-\beta), x-\beta
  9. (x-\alpha)(x-\beta)(x-\gamma), x-\alpha
  10. (x-\alpha)^4
  11. (x-\alpha)^3(x-\beta)
  12. (x-\alpha)^2(x-\beta)^2
  13. (x-\alpha)^2(x-\beta)(x-\gamma)
  14. (x-\alpha)(x-\beta)(x-\gamma)(x-\delta)

The possible lists of elementary divisors are as follows.

  1. x-\alpha, x-\alpha, x-\alpha, x-\alpha
  2. (x-\alpha)^2, (x-\alpha)^2
  3. (x-\alpha)^2, x-\alpha, x-\alpha
  4. x-\alpha, x-\beta, x-\alpha, x-\beta
  5. x-\alpha, x-\beta, x-\alpha, x-\alpha
  6. (x-\alpha)^3, x-\alpha
  7. (x-\alpha)^2, x-\beta, x-\alpha
  8. (x-\alpha)^2, x-\beta, x-\beta
  9. x-\alpha, x-\beta, x-\gamma, x-\alpha
  10. (x-\alpha)^4
  11. (x-\alpha)^3, x-\beta
  12. (x-\alpha)^2, (x-\beta)^2
  13. (x-\alpha)^2, x-\beta, x-\gamma
  14. x-\alpha, x-\beta, x-\gamma, x-\delta

The corresponding Jordan canonical forms are as follows.

  1. \begin{bmatrix} \alpha & 0 & 0 & 0 \\ 0 & \alpha & 0 & 0 \\ 0 & 0 & \alpha & 0 \\ 0 & 0 & 0 & \alpha \end{bmatrix}
  2. \begin{bmatrix} \alpha & 1 & 0 & 0 \\ 0 & \alpha & 0 & 0 \\ 0 & 0 & \alpha & 1 \\ 0 & 0 & 0 & \alpha \end{bmatrix}
  3. \begin{bmatrix} \alpha & 1 & 0 & 0 \\ 0 & \alpha & 0 & 0 \\ 0 & 0 & \alpha & 0 \\ 0 & 0 & 0 & \alpha \end{bmatrix}
  4. \begin{bmatrix} \alpha & 0 & 0 & 0 \\ 0 & \beta & 0 & 0 \\ 0 & 0 & \alpha & 0 \\ 0 & 0 & 0 & \beta \end{bmatrix}
  5. \begin{bmatrix} \alpha & 0 & 0 & 0 \\ 0 & \beta & 0 & 0 \\ 0 & 0 & \alpha & 0 \\ 0 & 0 & 0 & \alpha \end{bmatrix}
  6. \begin{bmatrix} \alpha & 1 & 0 & 0 \\ 0 & \alpha & 1 & 0 \\ 0 & 0 & \alpha & 0 \\ 0 & 0 & 0 & \alpha \end{bmatrix}
  7. \begin{bmatrix} \alpha & 1 & 0 & 0 \\ 0 & \alpha & 0 & 0 \\ 0 & 0 & \beta & 0 \\ 0 & 0 & 0 & \alpha \end{bmatrix}
  8. \begin{bmatrix} \alpha & 1 & 0 & 0 \\ 0 & \alpha & 0 & 0 \\ 0 & 0 & \beta & 0 \\ 0 & 0 & 0 & \beta \end{bmatrix}
  9. \begin{bmatrix} \alpha & 0 & 0 & 0 \\ 0 & \beta & 0 & 0 \\ 0 & 0 & \gamma & 0 \\ 0 & 0 & 0 & \alpha \end{bmatrix}
  10. \begin{bmatrix} \alpha & 1 & 0 & 0 \\ 0 & \alpha & 1 & 0 \\ 0 & 0 & \alpha & 1 \\ 0 & 0 & 0 & \alpha \end{bmatrix}
  11. \begin{bmatrix} \alpha & 1 & 0 & 0 \\ 0 & \alpha & 1 & 0 \\ 0 & 0 & \alpha & 0 \\ 0 & 0 & 0 & \beta \end{bmatrix}
  12. \begin{bmatrix} \alpha & 1 & 0 & 0 \\ 0 & \alpha & 0 & 0 \\ 0 & 0 & \beta & 1 \\ 0 & 0 & 0 & \beta \end{bmatrix}
  13. \begin{bmatrix} \alpha & 1 & 0 & 0 \\ 0 & \alpha & 0 & 0 \\ 0 & 0 & \beta & 0 \\ 0 & 0 & 0 & \gamma \end{bmatrix}
  14. \begin{bmatrix} \alpha & 0 & 0 & 0 \\ 0 & \beta & 0 & 0 \\ 0 & 0 & \gamma & 0 \\ 0 & 0 & 0 & \delta \end{bmatrix}
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Comments

  • Joe  On June 9, 2012 at 1:44 am

    Thanks for this post. It’s very helpful to me as I check my worked out Jordan exercises.

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