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](http://www.cscamm.umd.edu/tadmor/pub/linear-stability/GTZ%20spectral%20matrices%20LMA75.pdf)


$$\rho(A) \leq r(A) \leq \lVert A\rVert\_2 \leq 2 r(A).$$ 


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



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 $s(A) = 2|x|$. (This $A$ is essentially the most general $2\times 2$ matrix whose eigenvalues are both $0$.)


