Pairs of $k$-step reachability and $m$-step observability matrices

Let $V$ and $W$ be matrices of size $n \times pk$ and $q m \times n$, respectively. A necessary and sufficient condition is given for the existence of a triple $(A,B,C)$ such that $V$ a $k$-step reachability matrix of $(A,B)$ and $W$ an $m$-step observability matrix of $(A,C)$.

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