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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 existenceof a triple $(A,B,C)$ such that $V$ a $k$-step reachability matrix of $(A,B)$and $W$ an $m$-step observability matrix of $(A,C)$.

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