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If $P_1(s)$, $P_2(s)$, $Q_1(s)$, $Q_1(s)$ in R[s] to the power n x n, m x m, n x m, m x n respectively. How do we prove:

$((P_i)^{-1}(s))Q_i(s)$ is strictly proper for i = 1, 2 implies matrix($P_1 -Q_1; -Q_2 P_2$) has full row rank. (I am not sure how to write this in mathematical language: the above matrix has $P_1$ and -$Q_1$ in the first row and -$Q_2$ and $P_2$ in the second row.)

A matrix $A(s) = N(s) / d(s)$ where $N(s)$ is in $R[s]$ to the power $p \times q$ and $d(s) \neq 0$ is in $R[s]$ is strictly proper iff $deg(N_ij(s)) < deg(d(s))$ for all $i = 1,..,p$ and $j = 1,.., q$.

Thank you!

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