Let $A$ be a real square matrix of size $n \times n$. Is there an upper bound on the minimum spectral norm under diagonal similarity, i.e.,

$$ s(A) = \min_{D} \lVert D^{-1} A D\rVert_2, $$ where $D$ is a non-singular, diagonal real matrix. Also, is there are a relation between $s(A)$ and the spectral radius $\rho(A)$?

For the numerical radius $r(A) = \max_{\lVert x \rVert_2 = 1} \lVert \langle Ax, x\rangle \rVert$, it is

$$\rho(A) \leq r(A) \leq \lVert A\rVert_2 \leq 2 r(A).$$

I was hoping that it may be possible to minimize $r(D^{-1} A D)$.

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