[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.2.5

Show that matrices with distinct eigenvalues are dense in the space of all $n\times n$ matrices. (Use the Schur triangularisation)

 

Solution.  By the Schur triangularisation, for each matrix $A$, there exists a unitary $U$ such that $$\bex A=U\sex{\ba{ccc} \vLm_1&&*\\ &\ddots&\\ &&\vLm_s \ea},\quad \vLm_i=\sex{\ba{ccc} \lm_i&&*\\ &\ddots&\\ &&\lm_i \ea}_{n_i\times n_i}, \eex$$ with $\lm_1>\cdots>\lm_s$. For $\forall\ \ve>0$, we may replace the diagonal entries of $\vLm_i$ by $$\bex \lm_i+\frac{1}{ik} \eex$$ for $$\bex k>\max\sed{\frac{1}{n\ve},\max_{1\leq t<s}(\lm_t-\lm_{t+1})} \eex$$ to get a matrix $B_\ve$ with distinct eigenvalues with $\sen{A-B}_2<\ve$.