Hello,

I'll state the problem first and than I'll a little bit of motivation.

Lets be given regular matrix $M \in \mathbb{R}^{n\times n}$ and norm $||.||$ in $\mathbb{R}^{n}$. Define $$ U =\{ L\in \mathbb{R}^{n\times n}: \forall x\in \mathbb{R}^{n} \; ||LMx||=||x|| \}$$ (all those "norm fix" matrices for $M$) (sorry I have problems with curly brackets). Now the point is to find the best "norm fixing" matrix. I decided that the best one, call it $\bar{L}$, should satisfy: $$\inf_{L\in U} \; \; \sup_{||x||=1} \; ||LMx-Mx|| \; \; = \; \; \sup_{||x||=1} \; ||\bar{L}Mx-Mx||$$

The problem is to find the matrix $\bar{L}$ explicitly, not sure if $\bar{L}$ is unique but it exists. I'm most interested for p-norm with p equal 1 or 2.

Motivation: I was simulating some physical phenomena on computer. And the final equation basically boiled down to $x_{n+1} = Ax_n$. Often $x$ represents some quantity which is conserved. So I came up with this idea how to fix existing numerical scheme to conservative one (with least damage possible)