Show that matrices with distinct eigenvalues are dense in the space of all $n imes n$ matrices. (Use the Schur triangularisation)
Solution. By the Schur triangularisation, for each matrix $A$, there exists a unitary $U$ such that $$ex A=Usex{a{ccc} vLm_1&&*\ &ddots&\ &&vLm_s ea},quad vLm_i=sex{a{ccc} lm_i&&*\ &ddots&\ &&lm_i ea}_{n_i imes n_i}, eex$$ with $lm_1>cdots>lm_s$. For $forall ve>0$, we may replace the diagonal entries of $vLm_i$ by $$ex lm_i+frac{1}{ik} eex$$ for $$ex k>maxsed{frac{1}{nve},max_{1leq t<s}(lm_t-lm_{t+1})} eex$$ to get a matrix $B_ve$ with distinct eigenvalues with $sen{A-B}_2<ve$.