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From: crounse@norton.uucp (Great Rumpuscat)
Newsgroups: comp.ai.neural-nets,sci.math,sci.math.num-analysis
Subject: eigenvalue trivia/ref. wanted
Summary: circulant matrix
Message-ID: <42508@ucbvax.BERKELEY.EDU>
Date: 29 Jun 91 08:19:47 GMT
Sender: nobody@ucbvax.BERKELEY.EDU
Reply-To: crounse@norton.berkeley.edu
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A paper* I was just reading presents the following definition and
result:

A ``circulant'' is a matrix which has the same set of elements 
rotated by one position in each subsequent row, e.g.,

 / 1 2 3 4 \
 | 4 1 2 3 |
 | 3 4 1 2 |
 \ 2 3 4 1 /

A matrix which is a ``circulant'' has eigenvalues which are,
to a constant, the Inverse Discrete Fourier Transform of the row 
elements.


In the paper, this result is presented without reference.  This fact 
seems quite amazing (to me), and I would  expect that it must be
widely known in the proper circles.  If so, would someone please 
tell me where I could find a book or article which discusses such
matrices?

Thanks,
Ken Crounse 

* "Stability in Contractive Nonlinear Neural Networks," Douglas G. Kelly
  IEEE TrBioMedEng March 1990
,,,,,,,,,,,,,,,,,crounse@norton.berkeley.edu,,,,,,,,,,,,,,,,,,,,,,,
Kenneth R. Crounse,      -  UC Berkeley
King of                  -  (Rally Behind the Ridiculous)
Randomness and           -  Dept. of EECS
Chaos                    -  Nonlinear Electronics Laboratory