Suppose we have two matrices $A, B$ where $A \in \mathbb R^{n \times n}$ and $B \in \mathbb R^{n \times m}$. Further assume the rank of $[\lambda I - A, B]$ is $n$ for any $\lambda \in \mathbb C$ where $[\lambda I -A, B]$ denotes concatenating the columns.
I am wondering whether we can choose a continuous function $v\colon \mathbb C \to \mathbb C^{n+m}$ by $\lambda \mapsto v(\lambda)$ where $v(\lambda) \in \ker([\lambda I -A, B])$. We need $v$ not to be identically $0$.