A Real Nullstellensatz for Free Modules

Let $A$ be the algebra of all $n \times n$ matrices with entries from $\RR[x_1,\ldots,x_d]$ and let $G_1,\ldots,G_m,F \in A$. We will show that $F(a)v=0$ for every $a \in \RR^d$ and $v \in \RR^n$ such that $G_i(a)v=0$ for all $i$ if and only if $F$ belongs to the smallest real left ideal of $A$ which contains $G_1,\ldots,G_m$. Here a left ideal $J$ of $A$ is real if for every $H_1,\ldots,H_k \in A$ such that $H_1^T H_1+\ldots+H_k^T H_k \in J+J^T$ we have that $H_1,\ldots,H_k \in J$. We call this result the one-sided Real Nullstellensatz for matrix polynomials. We first prove by induction on $n$ that it holds when $G_1,\ldots,G_m,F$ have zeros everywhere except in the first row. This auxiliary result can be formulated as a Real Nullstellensatz for the free module $\RR[x_1,\ldots,x_d]^n$.