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Ordering of Eigensystem results

• To: mathgroup at smc.vnet.net
• Subject: [mg50791] Ordering of Eigensystem results
• From: AES/newspost <siegman at stanford.edu>
• Date: Tue, 21 Sep 2004 03:49:13 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

This post is just to report on a slightly odd ordering behavior that 
occurs for the results in a simple matrix Eigensystem calculation..

In physical terms I'm sending four complex-valued signals x1 thru x4 
through a sequence of cascaded lossless four-port scattering matrices 
which are chosen so that an input value x1=1 into the input end produces 
real and equal output values in all four output ports at the output end 
of the cascaded sequence.  (There are complex phase shifts in some of 
the individual scattering matrices within the cascaded system.)  

The sequence then has added to it a final (or equally well, an initial) 
scattering matrix which has unity amplitude transmission for channel 1 
and an amplitude transmission eps normally set equal to zero for the 
other three channels.

When I evaluate the Eigensystem for the resulting total system with eps 
= 0  the resulting eigenvalues are lambda = 1, 0, 0 and 0, which of 
course makes physical sense.  The lambda = 1 eigenvalue, however, is 
positioned as the _fourth_ and not the first eigenvalue in the list, 
whereas I've more commonly seen the largest magnitude eigenvalue as the 
first entry in the lists produced by Eigensystem.

If I give eps any small but finite value, the unity eigenvalue does in 
fact come out first.  Using eps = 0.0 also puts the unity eigenvalue 
first -- but using eps = 0 puts it last.

I'm not complaining about any of this behavior, or suggesting it's in 
any way erroneous.  But I guess I do have to keep in mind:  Although in 
many or even most Eigensystem calculations the eigenvalues will be 
arranged in order of decreasing magnitude, this is not always the case 
-- and indeed, I don't find anything in the Mathematica documentation 
that claims it will be.