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Let $A = (a_{ij})_{1 \leq i,j \leq N}$ and $B = (b_{ij})_{1 \leq i,j \leq N}$ be random matrices, with each $a_{ij}$ and $b_{ij}$ an independent random variable with continuous density function, zero mean and bounded variance. Let $\textbf{c}=(c_j)_{1 \leq j \leq N}$ be a random vector, whose entries are also independent and have a continuous, zero mean, bounded variance distribution.

Define the matrix $D$ by $$ D = \begin{pmatrix} A & 0 \\ BA &A\end{pmatrix}$$ and let $$\textbf{e} = \begin{pmatrix} \textbf{c} \\ B\textbf{c}\end{pmatrix}$$

Let $\textbf{q}_i$ be the generalized eigenvectors of $D$ for $1 \leq i \leq 2\cdot N$. What can we say about $\textbf{q}_i \cdot \textbf{e}$? In particular, is it possible to prove (or disprove) that $P \left( \bigcup_i \textbf{q}_i \cdot \textbf{e} \neq 0\right)=1$? That is, that $\textbf{e}$ is not orthogonal to any of the generalized eigenvectors of $D$ with probability one?

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