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Subject: Re: Finding the nearest matrix with real eigenvalues
Date: Fri, 3 Sep 2010 14:50:24 +0000 (UTC)
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In any case, I can now confirm that Roger's solution minimizes neither the Frobenius norm nor the spectral norm. The following gives a counter-example

A =   %given matrix

     1     1     0
    -1     0     0
     0     0     0


B1 =   %Roger's solution

    0.5000    0.0000         0
    0.0000    0.5000         0
         0         0         0


B =  %alternative to Roger's solution

    0.9000         0         0
   -2.0000   -0.1000         0
         0         0         0

Clearly both B and B1 have real eigenvalues, but one can directly verify that B achieves lower norms both Frobenius and spectral,

>> NormsB1 =[norm(A-B1,'fro'), norm(A-B1)]

NormsB1 =

    1.5811    1.5000

>> NormsB = [norm(A-B,'fro'), norm(A-B)],

NormsB =

    1.4213    1.1000