Suppose we have a real analytic family of matrices $A_t, B_t\in GL_n(\mathbb C)$, with $t\in \mathbb R$. Suppose that for $t\in (0,1)$ there is a common non-zero eigenvector $v_t\in \mathbb C^n\setminus 0$ such that $A_t(v_t)=B_t(v_t)=0$. Is it true that such a common (non-zero) vector exists for all $t\in \mathbb R$?