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From: "Bruno Luong" <b.luong@fogale.findmycountry>
Newsgroups: comp.soft-sys.matlab
Subject: Re: Finding the nearest matrix with real eigenvalues
Date: Fri, 3 Sep 2010 11:12:07 +0000 (UTC)
Organization: FOGALE nanotech
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"Matt J " <mattjacREMOVE@THISieee.spam> wrote in message <i5qi10$6oo$1@fred.mathworks.com>...
> "Bruno Luong" <b.luong@fogale.findmycountry> wrote in message <i5p8sn$mid$1@fred.mathworks.com>...

> 
> Bruno, although there are "floating" things in this norm, i.e., the scaling of the eigenvalues, the solution is invariant to them. 

But that's even more disturbing. There is a family of "norms" that depends on scaling and the input, whereas the solution does not depend on the scaling. This solution cannot be *characterized* as the minimum of a "clean" cost function. The minimum of the norm is a property. 

I'll caricature here, but I could also tell:

B = V'*real(D)*V, solution proposed by Roger, minimizes the cost function 

norm(Ap-B, 'fro') where Ap:=V'*real(D)*V, and [V D]=eig(A).

That claims of course does not have any intrinsic value and it's certainly true. In a way you are doing more or less like this, you define a norm that is floating.

Bruno