Compute the Jordan canonical form of a given matrix

Let A be a 2 \times 2 matrix with entries in \mathbb{Q} such that A^3 = I and A \neq I. Compute the rational canonical form of A and the Jordan canonical form of A over \mathbb{C}.


The minimal polynomial of A divides x^3-1 = (x-1)(x^2+x+1) and is not x-1, and so must be p(x) = x^2+x+1. Since A has dimension 2, p(x) is the list of invariant factors of A. So the rational canonical form of A is \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix}.

Now p(x) factors over \mathbb{C} as p(x) = (x - \frac{-1+i\sqrt{3}}{2})(x-\frac{-1-i\sqrt{3}}{2}). So the Jordan canonical form of A is \begin{bmatrix} \frac{-1+i\sqrt{3}}{2} & 0 \\ 0 & \frac{-1-i\sqrt{3}}{2} \end{bmatrix}.

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