Given a matrix $M$ of size $n\times n,$ we write its different eigenvalues by $x_1,x_2,\ldots,x_m$ with $m\leq n$ such that $|x_1|>|x_2|>|x_3|>\cdots|x_m|,$ and call $x_2\doteq ...
Given a matrix $M$, and it is perturbed by a matrix $E$. Denote the new perturbed matrix by $M'=M+\epsilon E$. I wish to find out the optimal $\epsilon^*$ that minimizes the spectral radius of $M'$. ...