# Minimize the spectral radius of a perturbed matrix

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 $M'$. Any suggestion to solve this problem (either numerical or analytic methods)?

Note that the task is to minimize $\rho(M')$, not the 2-norm $\|M'\|_2$. I tried to use Linear Matrix Inequality (LMI) to minimize $\|M'\|_2$, but corresponding $\epsilon$ does not minimize $\rho(M')$.

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