# 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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The transformations needed to achieve the Kalman decomposition are listed here. From the way you posed your question, it is not clear to me what your state matrix, input matrix, output matrix, and feedforward matrix are.

For the controllable canonical form, look here. Read down a ways and you will come to an explanation of the controllable canonical form.

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