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In mathematics, the spectral radius of a square matrix or a bounded linear operator is the supremum among the absolute values of the elements in its spectrum.

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Minimize spectral norm under diagonal similarity

You cannot get an upper bound in general in terms of the spectral radius $\rho(A)$. Counterexample: if $$ A = \begin{bmatrix} x & 1 \\ -x^2 & -x \end{bmatrix} $$ then $\rho(A) = 0$ and $s(A) = 2|x|$. …
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