Consider matrix algebra $\mathcal{M}_n(\mathbb{C})$ (acting on $n$ dimensional space $V$) and let $R$ be subring of matrices of $\mathcal{M}_n(\mathbb{C})$. Let $U$ be $n-1$ dimensional subspace of $n$ dimensional space $V$. Let given that all elements from $R$ have some eigenvector (which depends on element $r\in R$) inside $U$. Is it true that all elements from $R$ have same eigenvector?