Transformation of matrices for smaller condition number

I have matrices of very large condition numbers. I wonder if there are any techniques to transform the matrices into new matrices with smaller condition number, keeping the eigen properties the same as the original ones?

Solution 1:

If preserving the eigenvalues is necessary, then what you want to do is impossible. Take for instance the $\left\|\cdot \right\|_{2}$ norm, where it is easily shown that \begin{align} \mathrm{cond}_{2}(A) = \frac{|\lambda_{\max}|}{|\lambda_{\min}|} \end{align} From this we can see that the only way to reduce the condition number is to reduce $|\lambda_{\max}|$ or increase $|\lambda_{\min}|$. And as all norms on finite-dimensional vector spaces are equivalent, then you will get limited purchasing power attempting to use any other norm to measure condition number.