How to split the eigenvalues of a one-parameter family of matrices

摘要

We are concerned with families $F$ of $n imes n$-matrices $F(t)$ depending smoothly on the parameter $ t in mathbb{R}$. We survey results on the behaviour of eigenvalues of $F(t)$ for certain classes of matrices. We are especially interested in the question whether multiple eigenvalues can be avoided generically. In the set of families of symmetric matrices $F(t)$, for example, generically all eigenvalues of $F(t)$ are simple for all $t in mathbb{R}$. We consider a class of natural perturbations $widetilde{F}$ of a given matrix family $F$ such that $widetilde{F}$ lies in the generic class, i.e. $widetilde{F}$ avoids double eigenvalues `as far as possible'.