Let $W=(w\_{ij})\_{1 \leq i, j \leq N}$ and $\textbf{v}=(v\_j)\_{1 \leq j \leq N}$ be a random $N\times N$ matrix and N-vector, respectively, where all $w\_{ij}$ are jointly independent and have discrete distributions with non-zero variance. Likewise all $v\_j$ are jointly independent (and independent of $w\_{ij}$) and have discrete distributions with non-zero variance. Let $\textbf{q}\_1, \ldots, \textbf{q}\_N$ be the eigenvectors (or generalized eigenvectors) of $W$.


Can we show that as $N \rightarrow \infty$ the probability that
$P\left(\bigwedge\_i \,\textbf{v} \cdot \textbf{q}\_i \neq 0 \right) \rightarrow 1$, or else show that this is not the case?



Probability eigenvectors of discrete random matrix are orthogonal to discrete random vector
If there is no connection between the distributions for different $N$, it's not true.
Suppose $w\_{ij}$ and $v\_i$ all have Bernoulli distributions with parameter $1 - 2^{-N}$. Then with high probability, all entries of $\bf W$ and $\bf v$ are $1$, and $\bf v$ is orthogonal to all but one of the eigenvectors of $\bf W$.


