This question is a [follow up](https://mathoverflow.net/questions/306653/minimize-spectral-norm-under-diagonal-similarity).


Let $A$ be a real square matrix of size $n \times n$. How to determine the minimum spectral norm under diagonal similarity, i.e.,


$$
s(A) = \inf\_{D} \lVert D^{-1} A D\rVert\_2,
$$ 
where $D$ is a non-singular, diagonal real matrix. As it is unlikely to find an analytical upper bound, I would like to ask how $s(A)$ could be determined numericallly.



Numerical minimization spectral norm under diagonal similarity
Here is a better, more direct solution.


This problem can be cast as a *Generalized Eigenvalue Problem* as is shown by 
Boyd, El Ghaoui, Feron, and Balakrishnan on page 39 (§3.3) of [Linear Matrix Inequalities in System and Control Theory](https://web.stanford.edu/~boyd/lmibook/lmibook.pdf):


$$
s(A) = \inf \left\{\gamma \mid A^\*PA < \gamma^2 P \textrm{ for diagonal } P > 0
 \right\}
$$




---


**Previous answer**


Unfortunately behind a paywall, but following my own comment about chasing literature on matrix balancing, I found the following old paper that solves your problem (*EDIT*: As noted by Sebastian, this paper actually only provides a solution for a restricted case), not only for the operator norm, but for a variety of other norms.


T. Ström. [Minimization of norms and logarithmic norms by diagonal similarities.](https://link.springer.com/article/10.1007/BF02242378) Computing, March 1972, Volume 10, Issue 1–2, pp 1–7.


