# Calculating the Kalman decomposition of a matrix?

If we are given the matrices $$A = \begin{bmatrix} -2 & 3 & 4 & 1\\ 1 & 6 & 6 & 3\\ 5 & 6 & 6 & 4\\ 0 & -17 & -19 & -8 \end{bmatrix} \quad\text{and}\quad b = \begin{bmatrix} 0 \\ -1 \\ 0 \\ 1 \end{bmatrix},$$ then $$[b\ Ab\ \dots\ A^{(n-1)}b] = \begin{bmatrix} 0 & -2 & -4 & -6\\ -1 & -3 & -5 & -7\\ 0 & -2 & -4 & -6\\ 1 & 9 & 17 & 25 \end{bmatrix},$$ which has rank 2.

How do we find the Kalman decomposition and the controllability canonical form of the controllable part?

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