Compute the Jordan and rational canonical forms of a given matrix over QQ

Let A = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ -2 & -2 & 0 & 1 \\ -2 & 0 & -1 & -2 \end{bmatrix}. Compute the Jordan and rational canonical forms of A over \mathbb{Q}.


Let P = \dfrac{1}{4} \begin{bmatrix} 0 & -2 & 0 & 4 \\ -8 & -4 & -4 & -4 \\ 0 & 1 & 0 & 0 \\ -1 & -3 & 0 & 0 \end{bmatrix}. Evidently, P^{-1}AP = \begin{bmatrix} -1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} is in Jordan canonical form. Thus the elementary divisors of A are (x+1)^2, x-1, and x-1, so that the invariant factors of A are (x+1)^2(x-1) and x-1. So the rational canonical form of A is \begin{bmatrix} 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix}. (Computations carried out by WolframAlpha; see here.)

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