Norm preserving matrix fix

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)

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Edited your Latex, mostly on MO one needs double backslashes in \\{ and \\}, also put in spacing. Does the phrase "regular matrix" refer to a restriction of some kind? If your $M$ is invertible, and you are using $p=2,$ then all $L = O M^{-1}$ with $O \in O_n.$ – Will Jagy Jan 30 '12 at 22:33
Meanwhile, the symmetry group of the "unit sphere" with $p=1$ or $p=\infty$ is finite. – Will Jagy Jan 30 '12 at 22:36
Will: With regular matrix I meant invertible matrix(sorry i have still gaps in English terminology) and thanks for reply Survit: Why would I need ordering of $U$? $\inf_{L∈U}f(L)=\sup_{L∈U}−f(L)$ where $f$ is some function defined on $U$ – Tomas Skrivan Jan 31 '12 at 11:25