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We are given two square matrices $A$ and $B$ of the same size over the field of complex numbers and $\epsilon > 0$. Then it can be shown that there exist non-singular (even diagonalizable) matrices $A_1$ and $B_1$ such that $\|A-A_1\|<\epsilon, \|B-B_1\|<\epsilon. $ Could one show that $A_1$ and $B_1$ can be taken (in addition) such that the product $A_1B_1^{-1}$ has a simple spectrum?

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    Welcome on Math.SE ! You should show us what you have thought about and what you have tried to answer this question. This is essential to have an idea of what you already know and where are your difficulties. – Tom-Tom May 20 '14 at 08:39
  • What does a "simple spectrum" mean? – Algebraic Pavel May 22 '14 at 13:49

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