# How to choose two diagonal matrices minimizing the condition number

I have a matrix $A \in R^{n×n}$. I would like to choose two diagonal matrices $D_1,D_2 \in R^{n×n}$ such that $\text{cond}(D_1AD_2)$ should be minimal. How to provide such diagonal matrices?

## 2 Answers

This answer is a bit late. I think the following paper may help you

Braatz, Richard D., and Manfred Morari. "Minimizing the Euclidean condition number." SIAM Journal on Control and Optimization 32.6 (1994): 1763-1768.

One can read the Braatz, Morari's paper here (cf. Jiaqi's post)

https://www.researchgate.net/publication/238877232_Minimizing_the_Euclidean_Condition_Number

They consider results about the $$2$$ norm condition number.

The minimization about the $$\infty$$ norm condition number is standard and due to L. Bauer (cf. reference in the above one). Essentially, it consists in that follows

We can choose $$D_1,D_2$$ s.t. $$CN(D_1AD_2)=\rho(|A||A^{-1}|)$$, where $$|B|=[|b_{i,j}|]$$ (and it's the best choice).

Yet, a priori, the method does not seem very powerful; I wrote an example where

$$n=4, CN(A)\approx 3.10^{17}$$, and $$N(D_1AD_2)\approx 10^{17}$$.