Let $A=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, $B=\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$ we want to determine the eigenpairs of the general eigenvalue problem.
From the following properties we have:
$(\lambda,x) $ eigenpair of $(A,B) \iff (A-\lambda B)x=0$ , $x\ne 0$.
Then we have $P(\lambda)=-\lambda \Rightarrow \sigma(A,B)=\{ 0,\infty\}$
My question is how to determine the eigenvector in the case of $\lambda=\infty$ ?