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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.


Susan Blackford 2000-11-20