On the supremum norm of matrices
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Let $D=diag (d_{ii}) in M_n(mathbb R)$ be a diagonal matrix and $Ein M_n(mathbb R)$ be such that $||E||_infty < min _{ine j} Bigg|dfrac{d_{ii}-d_{jj}}{2}Bigg|$ . Then how to show that there is an ordering of the eigenvalues of $D+E$ as ${mu_1,...,mu_n}$ such that $|d_{ii}-mu_i|<||E||_infty $ ? I think I have to apply Gershgorin circle theorem, but I'm not quite sure how. Please help NOTE: Here $||E||_infty :=sup_{||x||_infty=1}||Ex||_infty$
linear-algebra matrices eigenvalues-eigenvectors norm gershgorin-sets
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edited Jan 11 at 6:26
user5