Generalized Non-Hermitian Eigenproblems
  J. Demmel

We assume that $A$ and $B$ are general $n$ by $n$ matrices. We call $A - \lambda B$ a matrix pencil, or pencil for short. The most common case, and the one we will deal with first, is the regular case, which occurs when $A$ and $B$ are square and the characteristic polynomial $p(\lambda) = {\rm det}(\lambda B-A)$ is not zero for all $\lambda$.This is equivalent to assuming that there are $n$ eigenvalues (finite or infinite) and that they are continuous functions of $A$ and $B$, i.e., that small changes in $A$ and $B$ cause small changes in the eigenvalues (this requires an appropriate definition for the case of infinite eigenvalues).

We will deal with the singular case at the end of this section.