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Subject: Re: Finding the nearest matrix with real eigenvalues
Date: Fri, 3 Sep 2010 22:01:09 +0000 (UTC)
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"Matt J " <mattjacREMOVE@THISieee.spam> wrote in message <i5r1vg$mg5$1@fred.mathworks.com>...
> In any case, I can now confirm that Roger's solution minimizes neither the Frobenius norm nor the spectral norm. .......
- - - - - - - - -
  Yes, Matt, I discovered that this morning while working with 2 x 2 real matrices.  In this case it is possible to derive an explicit expression for the matrix with real eigenvalues for which the difference from the original matrix is minimum under the Frobenius norm.  The results do not look at all like my earlier method which latter always gives a poorer answer.  As your results show, this is also the case for 3 x 3 matrices.  I haven't yet been able to discover a generalization of this 2 x 2 solution to larger matrices.

Roger Stafford